diff --git a/ArkLib.lean b/ArkLib.lean index 40f60595b6..8d835d6c6f 100644 --- a/ArkLib.lean +++ b/ArkLib.lean @@ -255,6 +255,19 @@ import ArkLib.ProofSystem.Whir.MutualCorrAgreement import ArkLib.ProofSystem.Whir.OutofDomainSmpl import ArkLib.ProofSystem.Whir.ProximityGen import ArkLib.ProofSystem.Whir.RBRSoundness +import ArkLib.Scratch.KzgVacuity.GgmAdaptive +import ArkLib.Scratch.KzgVacuity.GgmArkLibTransport +import ArkLib.Scratch.KzgVacuity.GgmCandidate +import ArkLib.Scratch.KzgVacuity.GgmDegreeDischarge +import ArkLib.Scratch.KzgVacuity.GgmDegreeInvariant +import ArkLib.Scratch.KzgVacuity.GgmEmbed +import ArkLib.Scratch.KzgVacuity.GgmEndToEnd +import ArkLib.Scratch.KzgVacuity.GgmProbThreading +import ArkLib.Scratch.KzgVacuity.GgmRandomEncoding +import ArkLib.Scratch.KzgVacuity.GgmShoup +import ArkLib.Scratch.KzgVacuity.GgmShoupEmbed +import ArkLib.Scratch.KzgVacuity.KzgVacuity +import ArkLib.Scratch.KzgVacuity.RepairSurvives import ArkLib.ToCompPoly.Univariate.Basic import ArkLib.ToCompPoly.Univariate.Lagrange import ArkLib.ToMathlib.BigOperators.Fin diff --git a/ArkLib/Commitments/Functional/KZG/Binding.lean b/ArkLib/Commitments/Functional/KZG/Binding.lean index 6a56e565d6..a4d1ef01f9 100644 --- a/ArkLib/Commitments/Functional/KZG/Binding.lean +++ b/ArkLib/Commitments/Functional/KZG/Binding.lean @@ -739,22 +739,32 @@ lemma t_sdh_error_bound {n : ℕ} {AuxState : Type} [SampleableType G₁] adversary) omit [DecidableEq G₁] in -/-- The KZG scheme satisfies evaluation binding provided `t`-SDH holds. -/ -theorem binding {g₁ : G₁} {g₂ : G₂} (hg₁ : g₁ ≠ 1) - (hpair : pairing g₁ g₂ ≠ 0) [SampleableType G₁] (tSdhError : ℝ≥0) - (htSdh : Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) - (g₂ := g₂) n tSdhError) : - Commitment.binding (init := pure ∅) (impl := randomOracle) - (kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing)) tSdhError := by +/-- **Extraction-shaped evaluation binding for KZG.** Every evaluation-binding adversary +`adversary` *yields*, as explicit data — the reduction `bindingReduction … adversary` — a +`t`-SDH adversary whose success probability upper-bounds `adversary`'s binding advantage. + +This is the full constructive content of `binding`, stated *without* the +universally-quantified `Groups.tSdhAssumption` hypothesis. It is therefore immune to the +fact that `tSdhAssumption … error` is false below error `1` for an unrestricted adversary +class (a `Classical.choice` adversary can read the trapdoor out of the verifier SRS leg): +there is no assumption `Prop` here for such an adversary to inhabit. The bound relates two +concrete probabilities of *this* adversary and *its* reduction. It is a genuine reduction +statement at every parameter, although the right-hand probability may of course be large for a +particular adversary. `binding` below is the immediate corollary obtained by bounding the +reduction's own success probability via the assumption. -/ +theorem binding_reduces_to_tSdh {g₁ : G₁} {g₂ : G₂} (hg₁ : g₁ ≠ 1) + (hpair : pairing g₁ g₂ ≠ 0) [SampleableType G₁] (AuxState : Type) + (adversary : KzgBindingAdversary p G₁ G₂ n unifSpec AuxState) : + Commitment.bindingExperiment (init := pure ∅) (impl := randomOracle) + (kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing)) AuxState adversary + ≤ Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) n + (bindingReduction (g₁ := g₁) (g₂ := g₂) (pairing := pairing) AuxState adversary) := by letI scheme := kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing) - simp only [Commitment.binding] - intro AuxState adversary letI game := Commitment.bindingGame (init := pure ∅) (impl := randomOracle) (AuxState := AuxState) (scheme := scheme) (adversary := adversary) letI game_ext := bindingGameExt (g₁ := g₁) (g₂ := g₂) AuxState adversary scheme - change Pr[Commitment.bindingCondition (Data := Fin (n + 1) → ZMod p) | game] ≤ tSdhError - exact - calc Pr[Commitment.bindingCondition (Data := Fin (n + 1) → ZMod p) | game] + change Pr[Commitment.bindingCondition (Data := Fin (n + 1) → ZMod p) | game] ≤ _ + calc Pr[Commitment.bindingCondition (Data := Fin (n + 1) → ZMod p) | game] _ = Pr[bindingCondExt (p := p) (n := n) | game_ext] := binding_game_ext_eq_binding_game (pairing := pairing) adversary _ ≤ Pr[(Groups.tSdhCondition (p := p) (g₁ := g₁)) ∘ mapBindingToTsdh (p := p) @@ -766,8 +776,26 @@ theorem binding {g₁ : G₁} {g₂ : G₂} (hg₁ : g₁ ≠ 1) _ = Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) n (bindingReduction (g₁ := g₁) (g₂ := g₂) (pairing := pairing) AuxState adversary) := t_sdh_game_eq (g₁ := g₁) (g₂ := g₂) (pairing := pairing) adversary - _ ≤ tSdhError := t_sdh_error_bound (g₁ := g₁) (g₂ := g₂) (pairing := pairing) - tSdhError htSdh adversary + +omit [DecidableEq G₁] in +/-- The KZG scheme satisfies evaluation binding provided `t`-SDH holds. + +This is an immediate corollary of `binding_reduces_to_tSdh`: for each adversary, compose +the (unconditional) reduction bound with the assumption's bound on the reduction's own +success probability. For the current, unrestricted `Groups.tSdhAssumption` type this corollary is +vacuous below error `1`. A future restricted assumption would require a correspondingly changed +corollary that proves `bindingReduction … adversary` belongs to the restricted class. The +assumption-free reduction statement lives in `binding_reduces_to_tSdh`. -/ +theorem binding {g₁ : G₁} {g₂ : G₂} (hg₁ : g₁ ≠ 1) + (hpair : pairing g₁ g₂ ≠ 0) [SampleableType G₁] (tSdhError : ℝ≥0) + (htSdh : Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) + (g₂ := g₂) n tSdhError) : + Commitment.binding (init := pure ∅) (impl := randomOracle) + (kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing)) tSdhError := by + simp only [Commitment.binding] + intro AuxState adversary + exact (binding_reduces_to_tSdh (pairing := pairing) hg₁ hpair AuxState adversary).trans + (t_sdh_error_bound (g₁ := g₁) (g₂ := g₂) (pairing := pairing) tSdhError htSdh adversary) end Binding diff --git a/ArkLib/Scratch/KzgVacuity/GgmAdaptive.lean b/ArkLib/Scratch/KzgVacuity/GgmAdaptive.lean new file mode 100644 index 0000000000..38ae4053cc --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmAdaptive.lean @@ -0,0 +1,426 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Scratch.KzgVacuity.GgmCandidate + +/-! +# Adaptive generic-group $t$-SDH bound + +The static bound of `GgmCandidate` (the generic group model [Sho97], [Mau05] $q = 0$ fragment) +is pushed here to $q$ adaptive oracle queries, following the Boneh–Boyen $t$-SDH argument [BB04]. +Every committed generic adversary wins $t$-SDH on $\le (D+1)/(p-1)$ of trapdoors when $q = 0$; this +file builds a deterministic adaptive object: a strategy making $q$ oracle queries — group +operations (linear combinations) and equality tests between opaque handles — before committing its +output. The oracle answers equality symbolically (formal polynomial equality), never revealing +$\tau$. + +The oracle has no pairing move: ArkLib's `tSdhAdversary D` maps +`Vector G₁ (D+1) × Vector G₂ 2 → … (Option (ZMod p × G₁))`, must output a $G_1$ element, and is +granted no pairing map $e : G_1 \times G_2 \to G_t$. Every handle it forms is a +$\mathbb{Z}/p$-linear combination of the seed $\{1, X, \dots, X^D\}$, degree $\le D$ (never a +product). This is the linear-oracle model on the critical path ($\delta = D$). + +The Shoup argument is mechanized at the counting/set level (no probability monad, the same idiom +as the static file): + +1. The generic-group oracle: handles are $\mathbb{N}$ indices into a table of formal polynomials in + $(\mathbb{Z}/p)[X]$, seeded with the SRS $1, X, \dots, X^D$ ($G_1$) and $1, X$ ($G_2$). Moves + append linear combinations of existing handles; equality is answered by an abstract `AnswerFn`. + +2. Identical-until-bad (the crux, proved not assumed): run the adversary against two answer + functions. If they agree on every pair actually queried in the first run, the runs coincide + step-for-step (`runAux_congr_of_agree`, by induction on fuel). The real oracle (equality at + $\tau$) and the symbolic oracle (formal equality) agree on a queried pair $(f_i, f_j)$ unless + $f_i \ne f_j$ formally but $f_i(\tau) = f_j(\tau)$ — that is, $\tau$ is a root of the nonzero + difference: the bad event. + +3. The bound: `realWinSet ⊆ symbolicWinSet ∪ badSet`. The symbolic set is bounded by the static + core (`GgmCandidate.card_winningPoints_le`, reused). The bad set is a union of + $\le \#\text{pairs}$ root-sets of nonzero polynomials of degree $\le \Delta$, bounded by + Schwartz–Zippel [Sch80], [Zip79]. Composing gives + $(D + 1 + \#\text{pairs} \cdot \Delta)/(p-1) \sim (q+D)^2 D / p$. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +* [Maurer, U., *Abstract Models of Computation in Cryptography*][Mau05] +* [Schwartz, J. T., *Fast Probabilistic Algorithms for Verification of Polynomial + Identities*][Sch80] +* [Zippel, R., *Probabilistic Algorithms for Sparse Polynomials*][Zip79] +-/ + +open Polynomial + +namespace GgmAdaptive + +open GgmCandidate + +variable {p : ℕ} [Fact (Nat.Prime p)] + +/-! ## Core 1 — the bad-set union-of-roots bound (Schwartz–Zippel, union form) + +Given a finite family of polynomials each of degree ≤ Δ, the set of field points that are a +root of *at least one* has cardinality ≤ (#polys)·Δ. (Mathlib defines the root multiset of the zero +polynomial to be empty.) This bounds the Shoup bad event once the +handle-difference polynomials are collected. -/ + +/-- The union of the root sets of a finite family of polynomials. -/ +noncomputable def rootUnion (ps : Finset ((ZMod p)[X])) : Finset (ZMod p) := + ps.biUnion (fun q => q.roots.toFinset) + +/-- **Union root bound.** If every polynomial in the family has degree ≤ Δ, +the union of their roots has card ≤ (#polys)·Δ. -/ +theorem card_rootUnion_le {ps : Finset ((ZMod p)[X])} {Δ : ℕ} + (hdeg : ∀ q ∈ ps, q.natDegree ≤ Δ) : + (rootUnion ps).card ≤ ps.card * Δ := by + classical + refine (Finset.card_biUnion_le).trans ?_ + calc ∑ q ∈ ps, (q.roots.toFinset).card + ≤ ∑ _q ∈ ps, Δ := by + refine Finset.sum_le_sum ?_ + intro q hq + refine (Multiset.toFinset_card_le q.roots).trans ?_ + exact (card_roots' q).trans (hdeg q hq) + _ = ps.card * Δ := by rw [Finset.sum_const, smul_eq_mul, mul_comm] + +/-! ## Core 2 — the generic-group oracle and its adaptive run + +Handles are `ℕ` indices into a `table : List (ZMod p)[X]` of formal polynomials, seeded with the +SRS handles. The adversary is `Strat`: given the history of equality-query answers (the *only* +τ-dependent input it receives in the generic model — Shoup 1997), it chooses the next `Move` or +commits an output `(offset c, handle index k)`. The oracle answers equality via an abstract +`AnswerFn`; the real oracle answers by evaluation at τ, the symbolic oracle by formal equality. -/ + +/-- How the oracle answers an equality query between the polynomials behind two handles. -/ +abbrev AnswerFn (p : ℕ) := (ZMod p)[X] → (ZMod p)[X] → Bool + +/-- A generic-group move. `lin` forms a `ZMod p`-linear combination of existing handles (group +add / negate / scalar-mul); `query` issues an equality test whose boolean answer feeds forward +into the adversary's next decision. + +There is **no pairing move**: ArkLib's `tSdhAdversary D` receives `Vector G₁ (D+1) × Vector G₂ 2` +and must output a `G₁` element, with **no pairing map** `e : G₁ × G₂ → Gₜ` in its interface. So +every handle it can form is a `ZMod p`-linear combination of the seed `{1, X, …, X^D}`, degree ≤ D +(never a product — no `≤ 2D` term). This is the linear-oracle model on the critical path; the +the separate pairing-degree peer model is not used by this `Move` type. -/ +inductive Move (p : ℕ) where + | lin : List (ZMod p × ℕ) → Move p + | query : ℕ → ℕ → Move p + +/-- A deterministic generic (adaptive) strategy: a decision function from the history of equality +answers to either a next move or a committed output `(offset, output-handle-index)`. It never +receives τ or a group element carrying τ — only the equality booleans. Randomized strategies are +not represented +by this type; handling them would require an additional random-tape/mixture argument. -/ +abbrev Strat (p : ℕ) := List Bool → Move p ⊕ (ZMod p × ℕ) + +/-- Oracle state: the handle table and the equality-answer history. -/ +structure St (p : ℕ) where + table : List ((ZMod p)[X]) + hist : List Bool + +/-- The polynomial produced by a linear-combination move: `Σ cᵢ · table[idxᵢ]`. -/ +noncomputable def combine (spec : List (ZMod p × ℕ)) (table : List ((ZMod p)[X])) : (ZMod p)[X] := + (spec.map (fun ci => C ci.1 * table.getD ci.2 0)).sum + +/-- **The adaptive generic-group run.** Fuel-bounded. Returns the committed output `(offset, +output polynomial)` together with the list of `(a,b)` polynomial pairs the adversary actually +queried for equality — the transcript we test the two oracles' agreement against. -/ +noncomputable def runAux (ans : AnswerFn p) (strat : Strat p) : + ℕ → St p → (ZMod p × (ZMod p)[X]) × List ((ZMod p)[X] × (ZMod p)[X]) + | 0, _ => ((0, 0), []) + | fuel + 1, st => + match strat st.hist with + | Sum.inr (c, k) => ((c, st.table.getD k 0), []) + | Sum.inl (Move.lin spec) => + runAux ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ + | Sum.inl (Move.query i j) => + let a := st.table.getD i 0 + let b := st.table.getD j 0 + let r := runAux ans strat fuel ⟨st.table, st.hist ++ [ans a b]⟩ + (r.1, (a, b) :: r.2) + +/-- The committed output `(offset, output polynomial)` of a run. -/ +noncomputable def runOutput (ans : AnswerFn p) (strat : Strat p) (fuel : ℕ) (st : St p) : + ZMod p × (ZMod p)[X] := + (runAux ans strat fuel st).1 + +/-- **IDENTICAL-UNTIL-BAD (the crux), PROVEN not assumed.** If two answer functions agree on every +pair actually queried in the `ans1` run, the two runs are *identical* — same output, same queried +transcript. Proof: induction on fuel; the run branches on the oracle only at `query` steps, and +agreement on the head query keeps the histories (hence all future decisions) in lockstep. -/ +theorem runAux_congr_of_agree {ans1 ans2 : AnswerFn p} (strat : Strat p) : + ∀ (fuel : ℕ) (st : St p), + (∀ ab ∈ (runAux ans1 strat fuel st).2, ans1 ab.1 ab.2 = ans2 ab.1 ab.2) → + runAux ans2 strat fuel st = runAux ans1 strat fuel st := by + intro fuel + induction fuel with + | zero => intro st _; rfl + | succ fuel ih => + intro st h + -- Case on the adversary's decision at the current history. + rcases hdec : strat st.hist with m | out + · cases m with + | lin spec => + -- lin move: no query; recurse on the extended table. + have e1 : runAux ans1 strat (fuel + 1) st + = runAux ans1 strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runAux, hdec] + have e2 : runAux ans2 strat (fuel + 1) st + = runAux ans2 strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runAux, hdec] + rw [e2, e1] + exact ih _ (by rw [e1] at h; exact h) + | query i j => + -- query move: the divergence point. + have e1 : runAux ans1 strat (fuel + 1) st + = ((runAux ans1 strat fuel + ⟨st.table, st.hist ++ [ans1 (st.table.getD i 0) (st.table.getD j 0)]⟩).1, + (st.table.getD i 0, st.table.getD j 0) :: + (runAux ans1 strat fuel + ⟨st.table, st.hist ++ [ans1 (st.table.getD i 0) (st.table.getD j 0)]⟩).2) := by + simp only [runAux, hdec] + have e2 : runAux ans2 strat (fuel + 1) st + = ((runAux ans2 strat fuel + ⟨st.table, st.hist ++ [ans2 (st.table.getD i 0) (st.table.getD j 0)]⟩).1, + (st.table.getD i 0, st.table.getD j 0) :: + (runAux ans2 strat fuel + ⟨st.table, st.hist ++ [ans2 (st.table.getD i 0) (st.table.getD j 0)]⟩).2) := by + simp only [runAux, hdec] + -- head-of-transcript agreement: ans1 a b = ans2 a b + have hhead : ans1 (st.table.getD i 0) (st.table.getD j 0) + = ans2 (st.table.getD i 0) (st.table.getD j 0) := + h (st.table.getD i 0, st.table.getD j 0) (by rw [e1]; simp) + rw [e2, e1, ← hhead] + have htail : ∀ ab ∈ (runAux ans1 strat fuel + ⟨st.table, st.hist ++ [ans1 (st.table.getD i 0) (st.table.getD j 0)]⟩).2, + ans1 ab.1 ab.2 = ans2 ab.1 ab.2 := by + intro ab hab + apply h; rw [e1]; simp only [List.mem_cons]; right; exact hab + rw [ih _ htail] + · -- output: base case, both stop identically. + have e1 : runAux ans1 strat (fuel + 1) st = ((out.1, st.table.getD out.2 0), []) := by + simp only [runAux, hdec] + have e2 : runAux ans2 strat (fuel + 1) st = ((out.1, st.table.getD out.2 0), []) := by + simp only [runAux, hdec] + rw [e2, e1] + +/-- The number of queried pairs in a run is bounded by the fuel: each query consumes one step. -/ +theorem runAux_queries_length_le (ans : AnswerFn p) (strat : Strat p) : + ∀ (fuel : ℕ) (st : St p), (runAux ans strat fuel st).2.length ≤ fuel := by + intro fuel + induction fuel with + | zero => intro st; simp [runAux] + | succ fuel ih => + intro st + rcases hdec : strat st.hist with m | out + · cases m with + | lin spec => + have e : runAux ans strat (fuel + 1) st + = runAux ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runAux, hdec] + rw [e]; exact (ih _).trans (Nat.le_succ _) + | query i j => + have e : runAux ans strat (fuel + 1) st + = ((runAux ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩).1, + (st.table.getD i 0, st.table.getD j 0) :: + (runAux ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩).2) := by + simp only [runAux, hdec] + rw [e]; simp only [List.length_cons] + exact Nat.succ_le_succ (ih _) + · have e : runAux ans strat (fuel + 1) st = ((out.1, st.table.getD out.2 0), []) := by + simp only [runAux, hdec] + rw [e]; simp + +/-! ## Core 3 — the symbolic / real oracles and the composed adaptive bound + +The symbolic oracle answers equality by *formal* polynomial equality (τ absent); the real oracle +answers by evaluation at τ. `runAux_congr_of_agree` couples them. -/ + +open Classical in +/-- Symbolic oracle: equality is *formal* polynomial equality, so the whole symbolic run is +τ-independent (the delayed-sampling simulator, Boneh–Boyen). -/ +noncomputable def symAns : AnswerFn p := fun f g => decide (f = g) + +/-- Real oracle at trapdoor `τ`: equality is evaluation equality at `τ`. -/ +noncomputable def realAns (τ : ZMod p) : AnswerFn p := fun f g => decide (f.eval τ = g.eval τ) + +variable (strat : Strat p) (st₀ : St p) (fuel : ℕ) + +/-- The τ-independent symbolic committed output `(offset, output polynomial)`. -/ +noncomputable def symOutput : ZMod p × (ZMod p)[X] := runOutput symAns strat fuel st₀ + +/-- The τ-independent list of equality-queried pairs in the symbolic run. -/ +noncomputable def symPairs : List ((ZMod p)[X] × (ZMod p)[X]) := (runAux symAns strat fuel st₀).2 + +/-- The bad-event polynomials: differences `a − b` of every *formally distinct* queried pair. +Each is nonzero (Shoup's nonzero-difference requirement). -/ +noncomputable def badPolys : Finset ((ZMod p)[X]) := + ((symPairs strat st₀ fuel).filter (fun ab => decide (ab.1 ≠ ab.2))).map + (fun ab => ab.1 - ab.2) |>.toFinset + +/-- The Shoup **bad set**: trapdoors on which two formally-distinct queried handles collide. -/ +noncomputable def badSet : Finset (ZMod p) := rootUnion (badPolys strat st₀ fuel) + +/-- **Agreement off the bad set.** For any `τ` outside the bad set, the real oracle at `τ` agrees +with the symbolic oracle on every pair the symbolic run queried. -/ +theorem realAns_agree_off_badSet {τ : ZMod p} (hτ : τ ∉ badSet strat st₀ fuel) : + ∀ ab ∈ (runAux symAns strat fuel st₀).2, + symAns ab.1 ab.2 = realAns τ ab.1 ab.2 := by + intro ab hab + simp only [symAns, realAns] + by_cases hfg : ab.1 = ab.2 + · simp [hfg] + · -- formally distinct: symbolic says "not equal"; show real agrees, i.e. eval differs. + have hne : ab.1 - ab.2 ≠ 0 := sub_ne_zero.mpr hfg + have hmem : ab.1 - ab.2 ∈ badPolys strat st₀ fuel := by + unfold badPolys + rw [List.mem_toFinset, List.mem_map] + refine ⟨ab, ?_, rfl⟩ + rw [List.mem_filter] + exact ⟨hab, by simp [hfg]⟩ + have hnotroot : τ ∉ (ab.1 - ab.2).roots.toFinset := by + intro hcontra + exact hτ (Finset.mem_biUnion.mpr ⟨_, hmem, hcontra⟩) + have hevalne : ab.1.eval τ ≠ ab.2.eval τ := by + intro hE + apply hnotroot + rw [Multiset.mem_toFinset, mem_roots hne] + simp only [IsRoot.def, eval_sub, hE, sub_self] + simp [hfg, hevalne] + +/-- **The real run equals the symbolic run off the bad set** — the mechanized identical-until-bad +step, specialized. Hence the real committed output coincides with the τ-independent symbolic one. -/ +theorem realOutput_eq_symOutput_off_badSet {τ : ZMod p} (hτ : τ ∉ badSet strat st₀ fuel) : + runOutput (realAns τ) strat fuel st₀ = symOutput strat st₀ fuel := by + simp only [runOutput, symOutput, runOutput] + rw [runAux_congr_of_agree strat fuel st₀ (realAns_agree_off_badSet strat st₀ fuel hτ)] + +/-! ### The composed adaptive bound + +`realWinSet` — the trapdoors on which the adaptive adversary wins t-SDH against the *real* oracle. +The win predicate is the τ+c≠0-guarded one reused verbatim from the static file (so Lean's total +`0⁻¹ = 0` cannot smuggle a spurious win). -/ + +/-- The trapdoors on which the adaptive adversary wins t-SDH against the real oracle at `τ`. -/ +noncomputable def realWinSet : Finset (ZMod p) := + nonzeroPoints.filter (fun τ => + τ + (runOutput (realAns τ) strat fuel st₀).1 ≠ 0 ∧ + (runOutput (realAns τ) strat fuel st₀).2.eval τ + = 1 / (τ + (runOutput (realAns τ) strat fuel st₀).1)) + +/-- **Identical-until-bad, at the set level.** Every real winning trapdoor either triggers the bad +event or is a static win of the τ-independent symbolic output `(c_sym, f_sym)` — Shoup's +`W₀ ⊆ W₁ ∪ F`, mechanized. -/ +theorem realWinSet_subset (D : ℕ) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) : + realWinSet strat st₀ fuel ⊆ + badSet strat st₀ fuel ∪ + GgmCandidate.winningPoints + (⟨(symOutput strat st₀ fuel).1, (symOutput strat st₀ fuel).2, hdeg_out⟩ : + GenericAdversary D p) := by + intro τ hτ + rw [realWinSet, Finset.mem_filter] at hτ + obtain ⟨hnz, hcond1, hcond2⟩ := hτ + by_cases hbad : τ ∈ badSet strat st₀ fuel + · exact Finset.mem_union_left _ hbad + · refine Finset.mem_union_right _ ?_ + have heq := realOutput_eq_symOutput_off_badSet strat st₀ fuel hbad + rw [heq] at hcond1 hcond2 + rw [GgmCandidate.winningPoints, Finset.mem_filter] + exact ⟨hnz, hcond1, hcond2⟩ + +/-- The number of bad-event polynomials is at most the fuel (one per equality query at most). -/ +theorem card_badPolys_le : (badPolys strat st₀ fuel).card ≤ fuel := by + unfold badPolys + refine (List.toFinset_card_le _).trans ?_ + rw [List.length_map] + exact (List.length_filter_le _ _).trans (runAux_queries_length_le symAns strat fuel st₀) + +/-- **THE ADAPTIVE GGM CARDINALITY BOUND.** For every deterministic strategy in this model making +≤ `fuel` oracle queries, the number of trapdoors on which it wins t-SDH is ≤ `fuel·Δ + (D+1)`: the +static Boneh–Boyen [BB04] root event `(D+1)` plus the Shoup [Sho97] collision event +`(#queries)·Δ`. -/ +theorem card_realWinSet_le (D Δ : ℕ) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_pairs : ∀ q ∈ badPolys strat st₀ fuel, q.natDegree ≤ Δ) : + (realWinSet strat st₀ fuel).card ≤ fuel * Δ + (D + 1) := by + classical + refine (Finset.card_le_card (realWinSet_subset strat st₀ fuel D hdeg_out)).trans ?_ + refine (Finset.card_union_le _ _).trans ?_ + have hbad : (badSet strat st₀ fuel).card ≤ fuel * Δ := + (card_rootUnion_le hdeg_pairs).trans + (Nat.mul_le_mul_right Δ (card_badPolys_le strat st₀ fuel)) + have hwin : (GgmCandidate.winningPoints + (⟨(symOutput strat st₀ fuel).1, (symOutput strat st₀ fuel).2, hdeg_out⟩ : + GenericAdversary D p)).card ≤ D + 1 := card_winningPoints_le _ + exact Nat.add_le_add hbad hwin + +/-- The adaptive success fraction: winning trapdoors over the `p−1` nonzero trapdoors. -/ +noncomputable def adaptiveExperiment : ℚ := (realWinSet strat st₀ fuel).card / (p - 1) + +/-- **THE ADAPTIVE GGM SECURITY BOUND (sorry-free).** Every deterministic strategy in this +explicit-equality model making ≤ `fuel` oracle queries wins on at most a +`(fuel·Δ + (D+1))/(p−1)` fraction of trapdoors. +This is the full Shoup / Boneh–Boyen shape — the static `(D+1)/(p−1)` root event plus the +`(#queries)·Δ/(p−1)` collision event — with `Δ = D` at faithful SRS degrees (the oracle has no +pairing, so every handle is a linear combination of the seed `{1, X, …, X^D}`, degree ≤ D, and +a difference of two such handles has degree ≤ D — never a product term). + +The two degree hypotheses are the SRS degree invariant (output handle is a G₁ element of degree +≤ D; queried-handle differences have degree ≤ Δ), true structurally for the faithful group-op +discipline. The identical-until-bad step (`realWinSet_subset`) is PROVEN by induction, not +assumed. -/ +theorem adaptive_ggm_sound (D Δ : ℕ) (hp : 2 ≤ p) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_pairs : ∀ q ∈ badPolys strat st₀ fuel, q.natDegree ≤ Δ) : + adaptiveExperiment strat st₀ fuel ≤ ((fuel * Δ + (D + 1) : ℕ) : ℚ) / (p - 1) := by + unfold adaptiveExperiment + have hnum : ((realWinSet strat st₀ fuel).card : ℚ) ≤ ((fuel * Δ + (D + 1) : ℕ) : ℚ) := by + exact_mod_cast card_realWinSet_le strat st₀ fuel D Δ hdeg_out hdeg_pairs + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by exact_mod_cast hp + linarith + gcongr + +omit [Fact (Nat.Prime p)] in +/-- **Non-vacuity of the adaptive bound.** Whenever `fuel·Δ + (D+1) < p − 1` the adaptive bound is +a genuine rational `< 1`: at cryptographic parameters (`p ≈ 2²⁵⁴`, `D ≈ 2²⁰`, `fuel = q ≈ 2⁶⁰`) +`fuel·Δ ≈ 2⁸⁰ ≪ p`, so the bound is `≈ 2⁻¹⁷⁴`. -/ +theorem adaptive_bound_lt_one (D Δ : ℕ) (hlt : fuel * Δ + (D + 1) < p - 1) (hp : 2 ≤ p) : + ((fuel * Δ + (D + 1) : ℕ) : ℚ) / (p - 1) < 1 := by + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by exact_mod_cast hp + linarith + rw [div_lt_one hden] + have h1 : ((fuel * Δ + (D + 1) : ℕ) : ℚ) < ((p - 1 : ℕ) : ℚ) := by exact_mod_cast hlt + have h2 : ((p - 1 : ℕ) : ℚ) = (p : ℚ) - 1 := by + have : (1 : ℕ) ≤ p := by omega + push_cast [Nat.cast_sub this]; ring + rw [h2] at h1; exact h1 + +/-- **The zero-fuel bound.** At `fuel = 0`, `runAux` returns its fixed fallback output `(0, 0)` +without consulting `strat` or `st₀`. Its success is bounded by the same numeric +`(D+1)/(p−1)` expression as the static theorem. This is only a boundary-case sanity check: it +does **not** embed or recover an arbitrary `GgmCandidate.GenericAdversary`. -/ +theorem adaptive_zero_fuel_bound (D : ℕ) (hp : 2 ≤ p) : + adaptiveExperiment strat st₀ 0 ≤ ((D + 1 : ℕ) : ℚ) / (p - 1) := by + have hout : (symOutput strat st₀ 0).2.natDegree ≤ D := by + simp [symOutput, runOutput, runAux] + have hpairs : ∀ q ∈ badPolys strat st₀ 0, q.natDegree ≤ 0 := by + intro q hq + simp [badPolys, symPairs, runAux] at hq + have := adaptive_ggm_sound strat st₀ 0 D 0 hp hout hpairs + simpa using this + +end GgmAdaptive + +#print axioms GgmAdaptive.runAux_congr_of_agree +#print axioms GgmAdaptive.card_realWinSet_le +#print axioms GgmAdaptive.adaptive_ggm_sound +#print axioms GgmAdaptive.adaptive_zero_fuel_bound diff --git a/ArkLib/Scratch/KzgVacuity/GgmArkLibTransport.lean b/ArkLib/Scratch/KzgVacuity/GgmArkLibTransport.lean new file mode 100644 index 0000000000..b0be288376 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmArkLibTransport.lean @@ -0,0 +1,191 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Scratch.KzgVacuity.GgmAdaptive +import ArkLib.Commitments.Functional.KZG.HardnessAssumptions + +/-! +# Transporting the GGM field-level t-SDH bound to the group-level game + +`GgmAdaptive` proves an adaptive generic-group-model (GGM) cardinality bound about a +*field-level* predicate: the generic adversary's committed polynomial satisfies +`f.eval τ = 1 / (τ + c)` at the sampled trapdoor `τ : ZMod p`. The Boneh–Boyen t-SDH hardness +game [BB04] that underlies the KZG polynomial commitment scheme [KZG10] instead scores a +*group-level* condition over a prime-order `G₁`: +$$\mathrm{tSdhCondition}(τ, c, h) := τ + c ≠ 0 ∧ h = g_1^{(1 / (τ + c)).\mathrm{val}}$$ +with `tSdhExperiment D A = Pr[tSdhCondition | tSdhGame D A]`. + +The two predicates live on opposite sides of the exponent encoding `a : ZMod p ↦ g ^ a.val`. +This file proves the encoding is injective — indeed bijective — in a prime-order group, hence the +two conditions are equivalent, hence the GGM winning-trapdoor set stated in group terms is +`GgmAdaptive.realWinSet`, and the adaptive bound `fuel·Δ + (D+1)` (and its `(…)/(p−1)` fraction) +transports verbatim to the group side. + +## Contents + +1. `gpow_val_injective` / `gpow_val_inj_iff` — injectivity of `a ↦ g ^ a.val` for `g` of + order `p`, derived from `Groups.gpow_div_eq` and `Groups.zmod_eq_zero_of_gpow_eq_one`; the + iff form. +2. `gpow_val_bijective` — bijectivity: surjectivity is `Groups.exists_zmod_power_of_generator` + (every element of a prime-order group is a `ZMod p` power of a nontrivial generator). +3. `tSdhCondition_iff_field` — for an output presented in encoded form `h = g ^ x.val`, + `Groups.tSdhCondition (τ, c, h)` holds iff `τ + c ≠ 0 ∧ x = 1 / (τ + c)` — exactly the + filter predicate of `GgmAdaptive.realWinSet` at `x = f.eval τ`. +4. `groupWinSet` / `groupWinSet_eq_realWinSet` / `field_bound_transports_to_group` — the set of + trapdoors on which the adaptive generic run's realized group element wins `tSdhCondition` + equals `realWinSet`; the cardinality bound and the rational fraction bound follow. + +## Named residual + +Connecting `field_bound_transports_to_group` to the literal inequality +`tSdhExperiment D A ≤ (fuel·Δ + D + 1) / (p−1)` requires threading VCVio's game monad — this is +probability-monad plumbing, with no new mathematics: + +* Generic-to-game embedding: for each `Strat p`, exhibit a `tSdhAdversary D`, + + Vector G₁ (D+1) × Vector G₂ 2 → + StateT unifSpec.QueryCache ProbComp (Option (ZMod p × G₁)) + + whose realized output on the SRS `PowerSrs.generate D τ` is + + ((runOutput (realAns τ) strat fuel st₀).1, + g₁ ^ ((runOutput (realAns τ) strat fuel st₀).2.eval τ).val) + + the generic-oracle-to-group simulation (equality queries answered by comparing real group + elements). +* Sampler semantics: `Pr[cond | sampleNonzeroZMod >>= deterministic] = + (nonzeroPoints.filter cond).card / (p−1)` — `probEvent` over `sampleNonzeroZMod` (a + `Fin (p−1)` uniform mapped by `i ↦ i+1`), plus the `ℚ → ℝ≥0∞` cast. + +Both are `OptionT ProbComp` bookkeeping. The condition-level identity proven here +(`groupWinSet_eq_realWinSet`) is exact, so the counting bound is about precisely the event +`tSdhExperiment` scores — nothing about the predicate remains to be aligned. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Kate, A., Zaverucha, G. M., and Goldberg, I., *Constant-Size Commitments to Polynomials and + Their Applications*][KZG10] +-/ + +open Polynomial + +namespace GgmArkLibTransport + +open GgmCandidate GgmAdaptive + +variable {p : ℕ} [Fact (Nat.Prime p)] +variable {G : Type} [Group G] [PrimeOrderWith G p] + +/-! ## 1. Injectivity of the exponent encoding `a : ZMod p ↦ g ^ a.val` -/ + +/-- **Injectivity.** For `g` of order `p`, the encoding `a : ZMod p ↦ g ^ a.val` is injective. +Derived from ArkLib's own prime-order lemmas: `g^a/g^b = g^(a−b)` (`Groups.gpow_div_eq`) and +`g^c = 1 → c = 0` (`Groups.zmod_eq_zero_of_gpow_eq_one`). -/ +theorem gpow_val_injective {g : G} (hord : orderOf g = p) : + Function.Injective (fun a : ZMod p => g ^ a.val) := by + intro a b hab + simp only at hab + have hdiv : g ^ a.val / g ^ b.val = (1 : G) := by rw [hab, div_self'] + rw [Groups.gpow_div_eq hord] at hdiv + exact sub_eq_zero.mp (Groups.zmod_eq_zero_of_gpow_eq_one hord hdiv) + +/-- **The iff.** `g ^ a.val = g ^ b.val ↔ a = b` for `g` of order `p`. -/ +theorem gpow_val_inj_iff {g : G} (hord : orderOf g = p) {a b : ZMod p} : + g ^ a.val = g ^ b.val ↔ a = b := + ⟨fun h => gpow_val_injective hord h, fun h => by rw [h]⟩ + +/-- **Bijectivity.** Injectivity above; surjectivity is exactly ArkLib's +`Groups.exists_zmod_power_of_generator` (every element of a prime-order group is a `ZMod p` +power of a nontrivial generator). The encoding is a bijection `ZMod p ≃ G` — the group carries +no more and no less information than the field of exponents. -/ +theorem gpow_val_bijective {g : G} (hpG : Nat.card G = p) (hg : g ≠ 1) + (hord : orderOf g = p) : + Function.Bijective (fun a : ZMod p => g ^ a.val) := + ⟨gpow_val_injective hord, fun x => by + obtain ⟨a, ha⟩ := Groups.exists_zmod_power_of_generator hpG hg hord x + exact ⟨a, ha.symm⟩⟩ + +/-! ## 2. The condition-level transport: `tSdhCondition` ↔ the field predicate -/ + +/-- **Condition transport.** For an output element presented in encoded form `g ^ x.val`, +ArkLib's group-level `tSdhCondition (τ, c, g ^ x.val)` holds **iff** the field-level condition +`τ + c ≠ 0 ∧ x = 1/(τ+c)` holds. Forward: injectivity. Backward: congruence. This is the +predicate `GgmAdaptive.realWinSet` filters by, at `x = f.eval τ`. -/ +theorem tSdhCondition_iff_field {g : G} (hord : orderOf g = p) (τ c x : ZMod p) : + Groups.tSdhCondition (g₁ := g) (τ, c, g ^ x.val) ↔ (τ + c ≠ 0 ∧ x = 1 / (τ + c)) := by + constructor + · rintro ⟨h1, h2⟩ + exact ⟨h1, gpow_val_injective hord h2⟩ + · rintro ⟨h1, h2⟩ + exact ⟨h1, by rw [h2]⟩ + +/-! ## 3. The set-level transport and the bound in group terms -/ + +/-- The trapdoors on which the adaptive generic run's realized **group** output wins ArkLib's +`tSdhCondition`: the committed offset is `(runOutput …).1` and the realized group element is the +encoding `g ^ (f τ).val` of the output polynomial's evaluation — exactly the element the +generic-model environment hands the t-SDH referee. (Decidability is supplied classically — +equality in the abstract group `G` carries no decision procedure.) -/ +noncomputable def groupWinSet (g : G) (strat : Strat p) (st₀ : St p) (fuel : ℕ) : + Finset (ZMod p) := + letI : DecidablePred (fun τ : ZMod p => + Groups.tSdhCondition (g₁ := g) + (τ, (runOutput (realAns τ) strat fuel st₀).1, + g ^ ((runOutput (realAns τ) strat fuel st₀).2.eval τ).val)) := + fun _ => Classical.dec _ + nonzeroPoints.filter (fun τ => + Groups.tSdhCondition (g₁ := g) + (τ, (runOutput (realAns τ) strat fuel st₀).1, + g ^ ((runOutput (realAns τ) strat fuel st₀).2.eval τ).val)) + +/-- **Set transport.** The group-level winning-trapdoor set IS the field-level one: +`groupWinSet g = GgmAdaptive.realWinSet`, pointwise by `tSdhCondition_iff_field`. -/ +theorem groupWinSet_eq_realWinSet {g : G} (hord : orderOf g = p) + (strat : Strat p) (st₀ : St p) (fuel : ℕ) : + groupWinSet g strat st₀ fuel = realWinSet strat st₀ fuel := by + classical + ext τ + simp only [groupWinSet, realWinSet, Finset.mem_filter, and_congr_right_iff] + intro _ _ + exact gpow_val_inj_iff hord + +/-- **THE TRANSPORTED ADAPTIVE BOUND (cardinality).** The number of trapdoors on which the +adaptive generic adversary's realized group element satisfies ArkLib's `tSdhCondition` is +≤ `fuel·Δ + (D+1)` — `GgmAdaptive.card_realWinSet_le`, now stated on the group side of the +encoding. -/ +theorem field_bound_transports_to_group {g : G} (hord : orderOf g = p) + (strat : Strat p) (st₀ : St p) (fuel : ℕ) (D Δ : ℕ) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_pairs : ∀ q ∈ badPolys strat st₀ fuel, q.natDegree ≤ Δ) : + (groupWinSet g strat st₀ fuel).card ≤ fuel * Δ + (D + 1) := by + rw [groupWinSet_eq_realWinSet hord strat st₀ fuel] + exact card_realWinSet_le strat st₀ fuel D Δ hdeg_out hdeg_pairs + +/-- **THE TRANSPORTED ADAPTIVE BOUND (fraction).** The fraction of the `p−1` nonzero trapdoors +on which the realized group element wins ArkLib's `tSdhCondition` is +≤ `(fuel·Δ + (D+1))/(p−1)` — `GgmAdaptive.adaptive_ggm_sound` on the group side. This rational +is the counting-level value of the probability `tSdhExperiment` assigns to the same condition +under uniform nonzero `τ` (see the NAMED RESIDUAL in the header for the `ProbComp` threading). -/ +theorem fraction_bound_transports_to_group {g : G} (hord : orderOf g = p) + (strat : Strat p) (st₀ : St p) (fuel : ℕ) (D Δ : ℕ) (hp : 2 ≤ p) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_pairs : ∀ q ∈ badPolys strat st₀ fuel, q.natDegree ≤ Δ) : + ((groupWinSet g strat st₀ fuel).card : ℚ) / (p - 1) + ≤ ((fuel * Δ + (D + 1) : ℕ) : ℚ) / (p - 1) := by + rw [groupWinSet_eq_realWinSet hord strat st₀ fuel] + exact adaptive_ggm_sound strat st₀ fuel D Δ hp hdeg_out hdeg_pairs + +/-! ## Axiom hygiene -/ + +#print axioms gpow_val_injective +#print axioms gpow_val_inj_iff +#print axioms gpow_val_bijective +#print axioms tSdhCondition_iff_field +#print axioms groupWinSet_eq_realWinSet +#print axioms field_bound_transports_to_group +#print axioms fraction_bound_transports_to_group + +end GgmArkLibTransport diff --git a/ArkLib/Scratch/KzgVacuity/GgmCandidate.lean b/ArkLib/Scratch/KzgVacuity/GgmCandidate.lean new file mode 100644 index 0000000000..41d50d4a39 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmCandidate.lean @@ -0,0 +1,177 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import Mathlib.Algebra.Polynomial.Roots +import Mathlib.Algebra.Field.ZMod +import Mathlib.Algebra.Order.Field.Basic +import Mathlib.Data.ZMod.Basic +import Mathlib.Data.Finset.Card + +/-! +# Static generic-group $t$-SDH soundness + +The `KzgVacuity` file shows that the concrete-group $t$-SDH assumption [BB04] is vacuous: ArkLib's +adversary receives the structured reference string as concrete group elements +`Vector G₁ (D+1) × Vector G₂ 2`, and from the verifier leg $g_2^{\tau}$ a +`Classical.choice`-definable adversary recovers the trapdoor and wins with probability $1$. + +This file establishes the *static* generic group model [Sho97], [Mau05] bound, the $q = 0$ +fragment against which the Boneh–Boyen $t$-SDH bound is proved. A generic adversary never sees +group elements as field data. We model the committed-generic adversary: without the trapdoor, it +commits to a challenge offset $c$ and a representation polynomial $f$ of degree $\le D$ over +$\mathbb{Z}/p$. The environment defines the output group element as $g_1^{f(\tau)}$; winning +requires $f(\tau) = 1/(\tau + c)$ at the environment's random $\tau$. + +Because $f$ is chosen with no $\tau$ in scope, there is nothing for `Classical.choice` to extract, +and the winning set of trapdoors is bounded by Schwartz–Zippel [Sch80], [Zip79]. `ggm_tSdh_sound` +proves the numeric bound $(D+1)/(p-1)$ for every committed adversary — including every +choice-definable one — so the concrete-group attack is dead in this model. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +* [Maurer, U., *Abstract Models of Computation in Cryptography*][Mau05] +* [Schwartz, J. T., *Fast Probabilistic Algorithms for Verification of Polynomial + Identities*][Sch80] +* [Zippel, R., *Probabilistic Algorithms for Sparse Polynomials*][Zip79] +-/ + +open Polynomial + +namespace GgmCandidate + +variable {p : ℕ} [Fact (Nat.Prime p)] + +/-! ## The winning polynomial and its Schwartz–Zippel degree bound -/ + +/-- The winning polynomial for a committed strategy `(c, f)`: +`w(X) = f(X) · (X + c) - 1`. A nonzero `τ` with `τ + c ≠ 0` wins iff `f(τ) = 1/(τ+c)`, i.e. iff +`w(τ) = 0`. -/ +noncomputable def winPoly (c : ZMod p) (f : (ZMod p)[X]) : (ZMod p)[X] := + f * (X + C c) - 1 + +/-- `winPoly` is never the zero polynomial: `f·(X+c) = 1` would force `deg(f·(X+c)) = 0`, but the +degree-1 factor `X + c` is nonzero over the field `ZMod p`, so the product has degree ≥ 1. -/ +lemma winPoly_ne_zero (c : ZMod p) (f : (ZMod p)[X]) : winPoly c f ≠ 0 := by + intro h + rw [winPoly, sub_eq_zero] at h -- h : f * (X + C c) = 1 + have hlin_ne : (X + C c : (ZMod p)[X]) ≠ 0 := (monic_X_add_C c).ne_zero + have hf_ne : f ≠ 0 := by + rintro rfl; rw [zero_mul] at h; exact zero_ne_one h + have hdeg := natDegree_mul hf_ne hlin_ne + rw [h, natDegree_one, natDegree_X_add_C] at hdeg + omega + +/-- `winPoly` has degree ≤ D + 1 when `f` has degree ≤ D. -/ +lemma winPoly_natDegree_le {D : ℕ} (c : ZMod p) {f : (ZMod p)[X]} (hf : f.natDegree ≤ D) : + (winPoly c f).natDegree ≤ D + 1 := by + unfold winPoly + refine (natDegree_sub_le _ _).trans ?_ + rw [natDegree_one] + refine max_le ?_ (by omega) + refine natDegree_mul_le.trans ?_ + have hlin : (X + C c : (ZMod p)[X]).natDegree = 1 := natDegree_X_add_C c + omega + +/-- **Schwartz–Zippel core.** The number of field points where a committed strategy `(c, f)` (with +`deg f ≤ D`) wins is bounded by the degree: `#roots(winPoly) ≤ D + 1`. -/ +lemma card_roots_winPoly_le {D : ℕ} (c : ZMod p) {f : (ZMod p)[X]} (hf : f.natDegree ≤ D) : + Multiset.card (winPoly c f).roots ≤ D + 1 := + (card_roots' (winPoly c f)).trans (winPoly_natDegree_le c hf) + +/-! ## The τ-free generic adversary and the counting experiment + +A committed generic adversary is a bare `(c, f)` — a challenge offset and a degree-≤D +representation polynomial — with **no trapdoor input**. This is the whole point: the type the +attack exploited (`Vector G₁ (D+1) × Vector G₂ 2 → …`, carrying `g₂^τ`) is gone, so +`Classical.choice` has no `∃ a, · = g^a` to invoke. -/ + +/-- A committed (static) generic t-SDH adversary: a challenge offset and a representation +polynomial of degree ≤ D, chosen independently of the trapdoor. -/ +structure GenericAdversary (D : ℕ) (p : ℕ) where + offset : ZMod p + repr : (ZMod p)[X] + degree_le : repr.natDegree ≤ D + +/-- The nonzero field elements — the support of ArkLib's trapdoor sampler `sampleNonzeroZMod`. -/ +noncomputable def nonzeroPoints : Finset (ZMod p) := + (Finset.univ : Finset (ZMod p)).erase 0 + +/-- The trapdoors on which the committed adversary `A` wins: nonzero `τ` with `τ + c ≠ 0` and +`f(τ) = 1/(τ+c)`. -/ +noncomputable def winningPoints {D : ℕ} (A : GenericAdversary D p) : Finset (ZMod p) := + nonzeroPoints.filter (fun τ => τ + A.offset ≠ 0 ∧ A.repr.eval τ = 1 / (τ + A.offset)) + +/-- Every winning `τ` is a root of `winPoly` (turning the rational win-condition into the +polynomial identity Schwartz–Zippel bounds). -/ +lemma winningPoints_subset_roots {D : ℕ} (A : GenericAdversary D p) : + ∀ τ ∈ winningPoints A, τ ∈ (winPoly A.offset A.repr).roots := by + intro τ hτ + rw [winningPoints, Finset.mem_filter] at hτ + obtain ⟨_, hne, heval⟩ := hτ + rw [mem_roots'] + refine ⟨winPoly_ne_zero A.offset A.repr, ?_⟩ + unfold winPoly + simp only [IsRoot.def, eval_sub, eval_mul, eval_add, eval_X, eval_C, eval_one] + rw [heval, one_div, inv_mul_cancel₀ hne, sub_self] + +/-- **The numeric GGM bound (counting form).** For EVERY committed generic adversary — including +every `Classical.choice`-definable one — the number of trapdoors on which it wins is ≤ D + 1. -/ +theorem card_winningPoints_le {D : ℕ} (A : GenericAdversary D p) : + (winningPoints A).card ≤ D + 1 := by + classical + have hsub : winningPoints A ⊆ (winPoly A.offset A.repr).roots.toFinset := by + intro τ hτ + rw [Multiset.mem_toFinset] + exact winningPoints_subset_roots A τ hτ + exact (Finset.card_le_card hsub).trans + ((Multiset.toFinset_card_le (m := (winPoly A.offset A.repr).roots)).trans + (card_roots_winPoly_le A.offset A.degree_le)) + +/-- The counting experiment: the fraction of nonzero trapdoors on which the committed adversary +wins. This is the exact success probability of the static generic adversary in the t-SDH game, +`τ` sampled uniformly from `sampleNonzeroZMod` (support = the `p - 1` nonzero residues). -/ +noncomputable def ggmExperiment {D : ℕ} (A : GenericAdversary D p) : ℚ := + (winningPoints A).card / (p - 1) + +/-! ## Survives-attack: the numeric bound holds for EVERY generic adversary -/ + +/-- **PROVEN-SURVIVES.** Every committed generic t-SDH adversary — over the FULL adversary type, +so including any `Classical.choice`/`Exists.choose`-defined one — wins on at most a `(D+1)/(p-1)` +fraction of trapdoors. There is no winning adversary at probability 1; the exact +trapdoor-extraction attack (`tauExtractingAdversary`) cannot even be typed here, because +`GenericAdversary` receives no group element and hence no `∃ a, · = g^a` for choice to invert. -/ +theorem ggm_tSdh_sound {D : ℕ} (A : GenericAdversary D p) (hp : 2 ≤ p) : + ggmExperiment A ≤ (D + 1 : ℚ) / (p - 1) := by + unfold ggmExperiment + have hmono : ((winningPoints A).card : ℚ) ≤ (D + 1 : ℚ) := by + exact_mod_cast card_winningPoints_le A + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by exact_mod_cast hp + linarith + gcongr + +omit [Fact (Nat.Prime p)] in +/-- **Non-vacuity is now built in.** For `p > D + 2` the bound `(D+1)/(p-1)` is a genuine rational +strictly below `1`: the assumption `∀ A, ggmExperiment A ≤ (D+1)/(p-1)` is TRUE (proved above, over +the whole type), not refutable, and its bound is nontrivial. Contrast `not_tSdhAssumption`, which +made the original assumption FALSE below `1`. -/ +theorem ggm_bound_lt_one {D : ℕ} (hp : D + 2 < p) : + ((D : ℚ) + 1) / (p - 1) < 1 := by + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by + have : (2 : ℕ) ≤ p := by omega + exact_mod_cast this + linarith + rw [div_lt_one hden] + have h1 : (D : ℚ) + 2 < (p : ℚ) := by exact_mod_cast hp + linarith + +end GgmCandidate + +#print axioms GgmCandidate.winPoly_ne_zero +#print axioms GgmCandidate.card_winningPoints_le +#print axioms GgmCandidate.ggm_tSdh_sound diff --git a/ArkLib/Scratch/KzgVacuity/GgmDegreeDischarge.lean b/ArkLib/Scratch/KzgVacuity/GgmDegreeDischarge.lean new file mode 100644 index 0000000000..265da357f2 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmDegreeDischarge.lean @@ -0,0 +1,313 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Scratch.KzgVacuity.GgmRandomEncoding +import ArkLib.Scratch.KzgVacuity.GgmDegreeInvariant + +/-! +# Degree discharge for the linear generic-group oracle + +`GgmAdaptive` (adaptive bound) and `GgmRandomEncoding` (all-pairs bound) consume the SRS degree +invariant as external hypotheses (`hdeg_out`, `hdeg_pairs`, `hdeg_handles`): the committed output +polynomial and every handle polynomial have `natDegree ≤ D`. Since the oracle is purely linear +(`Move.lin` only — no pairing move; ArkLib's `tSdhAdversary` is granted no pairing map), those +facts are true by construction. This file proves them, by induction on the actual run recursion +(`runAux` / `runTable`, not the separate `GgmDegreeInvariant.buildPaired` peer model): + +* `natDegree_combine_le` — a `lin` move's linear combination + $\sum_i C(c_i) \cdot \mathrm{table}[k_i]$ over a degree-$\le D$ table stays $\le D$ + (`natDegree_add_le` is a max-bound; `natDegree_C_mul_le` + absorbs the scalar; a defaulted out-of-range read is $0$). +* `runTable_natDegree_le` / `handlePolys_natDegree_le` — every polynomial in the run's final handle + table (resp. the `insert 0` handle set) has `natDegree ≤ D`, by induction on fuel. +* `runAux_output_natDegree_le` / `symOutput_natDegree_le` — the committed output polynomial has + `natDegree ≤ D` (it is a defaulted table read at commit time). +* `badPolys_natDegree_le` — every Shoup [Sho97] bad-event polynomial (difference of a formally + distinct queried pair) has `natDegree ≤ D` (`natDegree_sub_le` is a max-bound — the linear oracle + pays $\delta = D$, never $2D$). +* `srsSt_table_natDegree_le` — the SRS seed $1, X, \dots, X^D, 1, X$ meets the bound ($1 \le D$ + because the $G_2$ handle $X$ has degree $1$). +* the `_of_run` corollaries — `hdeg_out_of_run` / `hdeg_pairs_of_run` / `hdeg_handles_of_run` in + exactly the shape the existing theorems consume, plus the composed hypothesis-free bounds + `adaptive_ggm_sound_of_run` / `adaptive_ggm_sound_srs` / `rand_encoding_bound_D_of_run` / + `rand_encoding_bound_srs_D_of_run`. The degree facts are now theorems about the actual oracle, + not assumptions a downstream caller must supply. + +Reused: `GgmDegreeInvariant.natDegree_getD_le` (the defaulted-read bound); everything else here +targets the real `runAux`/`runTable` recursion directly. + +## References + +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +-/ + +open Polynomial + +namespace GgmDegreeDischarge + +open GgmCandidate GgmAdaptive GgmRandomEncoding GgmDegreeInvariant + +variable {p : ℕ} [Fact (Nat.Prime p)] + +/-! ## § Combine — a `lin` move preserves the degree bound -/ + +/-- `combine` on a cons: peel one summand. -/ +lemma combine_cons (ci : ZMod p × ℕ) (cs : List (ZMod p × ℕ)) (table : List ((ZMod p)[X])) : + combine (ci :: cs) table = C ci.1 * table.getD ci.2 0 + combine cs table := by + simp [combine] + +/-- **A `ZMod p`-linear combination of degree-≤D handles has degree ≤ D.** Induction on the +coefficient list: `natDegree_add_le` is a MAX bound, `natDegree_C_mul_le` absorbs the scalar, +and a defaulted table read is either a genuine (bounded) entry or the zero polynomial. -/ +theorem natDegree_combine_le {table : List ((ZMod p)[X])} {D : ℕ} + (h : ∀ q ∈ table, q.natDegree ≤ D) (spec : List (ZMod p × ℕ)) : + (combine spec table).natDegree ≤ D := by + induction spec with + | nil => simp [combine] + | cons ci cs ih => + rw [combine_cons] + refine (natDegree_add_le _ _).trans (max_le ?_ ih) + exact (natDegree_C_mul_le ci.1 _).trans (natDegree_getD_le h ci.2) + +/-! ## § Table — the run's final handle table stays degree-bounded + +Induction on fuel over the REAL `runTable` recursion: a `lin` step appends a `combine` +(bounded by `natDegree_combine_le`), a `query` step leaves the table unchanged, and a commit +stops. -/ + +/-- **The degree invariant of the real handle table.** If every seed polynomial has +`natDegree ≤ D`, so does every polynomial in the run's final table — for ANY answer function +(the table extension never consults the oracle's answers' values, only the history length +through the strategy). -/ +theorem runTable_natDegree_le (ans : AnswerFn p) (strat : Strat p) {D : ℕ} : + ∀ (fuel : ℕ) (st : St p), (∀ q ∈ st.table, q.natDegree ≤ D) → + ∀ q ∈ runTable ans strat fuel st, q.natDegree ≤ D := by + intro fuel + induction fuel with + | zero => + intro st hst q hq + simp only [runTable] at hq + exact hst q hq + | succ fuel ih => + intro st hst q hq + rcases hdec : strat st.hist with m | out + · cases m with + | lin spec => + have e : runTable ans strat (fuel + 1) st + = runTable ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runTable, hdec] + rw [e] at hq + refine ih _ ?_ q hq + intro r hr + rcases List.mem_append.mp hr with h | h + · exact hst r h + · rw [List.mem_singleton.mp h] + exact natDegree_combine_le hst spec + | query i j => + have e : runTable ans strat (fuel + 1) st + = runTable ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ := by + simp only [runTable, hdec] + rw [e] at hq + exact ih ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ hst q hq + · have e : runTable ans strat (fuel + 1) st = st.table := by + simp only [runTable, hdec] + rw [e] at hq + exact hst q hq + +/-- **The handle-set degree invariant** — `runTable_natDegree_le` extended to +`handlePolys = insert 0 (runTable …).toFinset` (the zero/identity handle has degree 0). -/ +theorem handlePolys_natDegree_le (ans : AnswerFn p) (strat : Strat p) (fuel : ℕ) (st₀ : St p) + {D : ℕ} (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + ∀ q ∈ handlePolys ans strat fuel st₀, q.natDegree ≤ D := by + intro q hq + unfold handlePolys at hq + rcases Finset.mem_insert.mp hq with h | h + · rw [h]; simp + · exact runTable_natDegree_le ans strat fuel st₀ hseed q (List.mem_toFinset.mp h) + +/-! ## § Output — the committed output polynomial is degree-bounded -/ + +/-- **The output degree invariant, over the real `runAux`.** The committed output is a +defaulted table read at commit time (or `0` on fuel exhaustion); every intermediate table is +bounded, so the output is. Induction on fuel, for ANY answer function. -/ +theorem runAux_output_natDegree_le (ans : AnswerFn p) (strat : Strat p) {D : ℕ} : + ∀ (fuel : ℕ) (st : St p), (∀ q ∈ st.table, q.natDegree ≤ D) → + ((runAux ans strat fuel st).1.2).natDegree ≤ D := by + intro fuel + induction fuel with + | zero => + intro st _ + simp [runAux] + | succ fuel ih => + intro st hst + rcases hdec : strat st.hist with m | out + · cases m with + | lin spec => + have e : runAux ans strat (fuel + 1) st + = runAux ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runAux, hdec] + rw [e] + refine ih _ ?_ + intro r hr + rcases List.mem_append.mp hr with h | h + · exact hst r h + · rw [List.mem_singleton.mp h] + exact natDegree_combine_le hst spec + | query i j => + have e : runAux ans strat (fuel + 1) st + = ((runAux ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩).1, + (st.table.getD i 0, st.table.getD j 0) :: + (runAux ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩).2) := by + simp only [runAux, hdec] + rw [e] + exact ih _ hst + · have e : runAux ans strat (fuel + 1) st = ((out.1, st.table.getD out.2 0), []) := by + simp only [runAux, hdec] + rw [e] + exact natDegree_getD_le hst out.2 + +/-- **The symbolic committed output has degree ≤ D** — the `hdeg_out` fact, proven about the +real run. -/ +theorem symOutput_natDegree_le (strat : Strat p) (st₀ : St p) (fuel : ℕ) {D : ℕ} + (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + (symOutput strat st₀ fuel).2.natDegree ≤ D := by + simpa only [symOutput, runOutput] using + runAux_output_natDegree_le symAns strat fuel st₀ hseed + +/-! ## § Pairs — the Shoup bad-event polynomials are degree-bounded -/ + +/-- **Every bad-event polynomial has degree ≤ D** — the `hdeg_pairs` fact at `Δ = D`, proven +about the real run: each is a difference of two queried handles, each of which lives in the +final table (or is `0`) by `runAux_pairs_mem_runTable`, and `natDegree_sub_le` is a MAX +bound — the linear oracle pays δ = D, never 2D. -/ +theorem badPolys_natDegree_le (strat : Strat p) (st₀ : St p) (fuel : ℕ) {D : ℕ} + (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + ∀ q ∈ badPolys strat st₀ fuel, q.natDegree ≤ D := by + intro q hq + unfold badPolys at hq + rw [List.mem_toFinset, List.mem_map] at hq + obtain ⟨ab, habf, rfl⟩ := hq + rw [List.mem_filter] at habf + obtain ⟨hab, -⟩ := habf + have hab' : ab ∈ (runAux symAns strat fuel st₀).2 := hab + have hmem := runAux_pairs_mem_runTable symAns strat fuel st₀ ab hab' + have h1 : ab.1.natDegree ≤ D := by + rcases hmem.1 with h | h + · exact runTable_natDegree_le symAns strat fuel st₀ hseed ab.1 h + · rw [h]; simp + have h2 : ab.2.natDegree ≤ D := by + rcases hmem.2 with h | h + · exact runTable_natDegree_le symAns strat fuel st₀ hseed ab.2 h + · rw [h]; simp + exact (natDegree_sub_le ab.1 ab.2).trans (max_le h1 h2) + +/-! ## § Seed — the SRS table meets the bound -/ + +/-- The SRS-seeded table `1, X, …, X^D, 1, X` has every entry of degree ≤ D. Needs `1 ≤ D`: +the G₂ handle `X` has degree 1 (at `D = 0` the seed itself would break the bound). -/ +theorem srsSt_table_natDegree_le (D : ℕ) (hD : 1 ≤ D) : + ∀ q ∈ (srsSt (p := p) D).table, q.natDegree ≤ D := by + intro q hq + simp only [srsSt] at hq + rcases List.mem_append.mp hq with h | h + · obtain ⟨k, hk, rfl⟩ := List.mem_map.mp h + exact (natDegree_X_pow_le k).trans (Nat.lt_succ_iff.mp (List.mem_range.mp hk)) + · rcases List.mem_cons.mp h with rfl | h + · simp + · rw [List.mem_singleton.mp h] + exact natDegree_X_le.trans hD + +/-! ## § Discharge — the `hdeg_*` hypotheses as theorems about the real oracle + +Thin named sockets in EXACTLY the hypothesis shapes `GgmAdaptive.adaptive_ggm_sound` / +`card_realWinSet_le` (`hdeg_out`, `hdeg_pairs`) and `GgmRandomEncoding.rand_encoding_bound_D` +& friends (`hdeg_out`, `hdeg_handles`) consume, followed by the composed bounds with the +degree hypotheses GONE. -/ + +/-- `hdeg_out`, discharged: the symbolic output degree bound holds for the real run. -/ +theorem hdeg_out_of_run (strat : Strat p) (st₀ : St p) (fuel D : ℕ) + (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + (symOutput strat st₀ fuel).2.natDegree ≤ D := + symOutput_natDegree_le strat st₀ fuel hseed + +/-- `hdeg_pairs` (at `Δ = D`), discharged: the bad-event degree bound holds for the real +run. -/ +theorem hdeg_pairs_of_run (strat : Strat p) (st₀ : St p) (fuel D : ℕ) + (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + ∀ q ∈ badPolys strat st₀ fuel, q.natDegree ≤ D := + badPolys_natDegree_le strat st₀ fuel hseed + +/-- `hdeg_handles` (at δ = D), discharged: the handle-set degree bound holds for the real +run's table (`runTable`, not a peer model). -/ +theorem hdeg_handles_of_run (strat : Strat p) (st₀ : St p) (fuel D : ℕ) + (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + ∀ q ∈ handlePolys symAns strat fuel st₀, q.natDegree ≤ D := + handlePolys_natDegree_le symAns strat fuel st₀ hseed + +/-- `GgmAdaptive.card_realWinSet_le` with both degree hypotheses DISCHARGED (Δ = D): only the +seed-table bound remains, and that is a theorem at the SRS seeding (below). -/ +theorem card_realWinSet_le_of_run (strat : Strat p) (st₀ : St p) (fuel D : ℕ) + (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + (realWinSet strat st₀ fuel).card ≤ fuel * D + (D + 1) := + card_realWinSet_le strat st₀ fuel D D + (symOutput_natDegree_le strat st₀ fuel hseed) + (badPolys_natDegree_le strat st₀ fuel hseed) + +/-- **`GgmAdaptive.adaptive_ggm_sound` with the degree hypotheses DISCHARGED** (Δ = D). -/ +theorem adaptive_ggm_sound_of_run (strat : Strat p) (st₀ : St p) (fuel D : ℕ) (hp : 2 ≤ p) + (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + adaptiveExperiment strat st₀ fuel ≤ ((fuel * D + (D + 1) : ℕ) : ℚ) / (p - 1) := + adaptive_ggm_sound strat st₀ fuel D D hp + (symOutput_natDegree_le strat st₀ fuel hseed) + (badPolys_natDegree_le strat st₀ fuel hseed) + +/-- **The adaptive bound at the SRS seeding, hypothesis-free** (`1 ≤ D` and `2 ≤ p` only): +every deterministic `Strat` against the SRS-seeded linear oracle wins on at most a +`(fuel·D + (D+1))/(p−1)` fraction of trapdoors — no degree assumption left. -/ +theorem adaptive_ggm_sound_srs (strat : Strat p) (fuel D : ℕ) (hD : 1 ≤ D) (hp : 2 ≤ p) : + adaptiveExperiment strat (srsSt D) fuel ≤ ((fuel * D + (D + 1) : ℕ) : ℚ) / (p - 1) := + adaptive_ggm_sound_of_run strat (srsSt D) fuel D hp (srsSt_table_natDegree_le D hD) + +/-- **`GgmRandomEncoding.rand_encoding_bound_D` with the degree hypotheses DISCHARGED**. -/ +theorem rand_encoding_bound_D_of_run (strat : Strat p) (st₀ : St p) (fuel D n : ℕ) + (hp : 2 ≤ p) (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) + (hn : st₀.table.length + fuel + 1 ≤ n) : + adaptiveExperiment strat st₀ fuel ≤ ((n.choose 2 * D + (D + 1) : ℕ) : ℚ) / (p - 1) := + rand_encoding_bound_D strat st₀ fuel D n hp + (symOutput_natDegree_le strat st₀ fuel hseed) + (handlePolys_natDegree_le symAns strat fuel st₀ hseed) hn + +/-- **The all-pairs δ = D bound at the SRS seeding, hypothesis-free** (`1 ≤ D`, +`2 ≤ p`): the concrete `(C(fuel+D+4, 2)·D + (D+1))/(p−1)` Shoup number with every degree +fact a theorem about the actual oracle. -/ +theorem rand_encoding_bound_srs_D_of_run (strat : Strat p) (fuel D : ℕ) (hD : 1 ≤ D) + (hp : 2 ≤ p) : + adaptiveExperiment strat (srsSt D) fuel ≤ + (((fuel + D + 4).choose 2 * D + (D + 1) : ℕ) : ℚ) / (p - 1) := + rand_encoding_bound_srs_D strat fuel D hp + (symOutput_natDegree_le strat (srsSt D) fuel (srsSt_table_natDegree_le D hD)) + (handlePolys_natDegree_le symAns strat fuel (srsSt D) (srsSt_table_natDegree_le D hD)) + +end GgmDegreeDischarge + +-- Axiom receipts: every theorem is sorry-free on the standard three axioms. +#print axioms GgmDegreeDischarge.natDegree_combine_le +#print axioms GgmDegreeDischarge.runTable_natDegree_le +#print axioms GgmDegreeDischarge.handlePolys_natDegree_le +#print axioms GgmDegreeDischarge.runAux_output_natDegree_le +#print axioms GgmDegreeDischarge.symOutput_natDegree_le +#print axioms GgmDegreeDischarge.badPolys_natDegree_le +#print axioms GgmDegreeDischarge.srsSt_table_natDegree_le +#print axioms GgmDegreeDischarge.hdeg_out_of_run +#print axioms GgmDegreeDischarge.hdeg_pairs_of_run +#print axioms GgmDegreeDischarge.hdeg_handles_of_run +#print axioms GgmDegreeDischarge.card_realWinSet_le_of_run +#print axioms GgmDegreeDischarge.adaptive_ggm_sound_of_run +#print axioms GgmDegreeDischarge.adaptive_ggm_sound_srs +#print axioms GgmDegreeDischarge.rand_encoding_bound_D_of_run +#print axioms GgmDegreeDischarge.rand_encoding_bound_srs_D_of_run diff --git a/ArkLib/Scratch/KzgVacuity/GgmDegreeInvariant.lean b/ArkLib/Scratch/KzgVacuity/GgmDegreeInvariant.lean new file mode 100644 index 0000000000..19baf8941c --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmDegreeInvariant.lean @@ -0,0 +1,301 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import Mathlib + +/-! +# Structural degree invariant for the generic-group handle table + +The `GgmAdaptive` file proves the adaptive generic group model [Sho97] $t$-SDH bound under two +external degree hypotheses (`hdeg_out`, `hdeg_pairs`): the polynomials in the oracle's handle +table have bounded `natDegree`. This file makes that fact structural: a `TableOp` inductive +mirroring the oracle's table-extension moves (SRS seed / linear combination / product), a +`buildTable` interpreter, and degree invariants proved by induction on the op list. + +The bounds — each proved below, none assumed: + +* `degree_invariant_linComb` — $B = D$ when no product op occurs. The seed powers $X^k$ + ($k \le D$) meet the bound exactly, and a linear combination degrades to the max of its + operands' degrees (`Polynomial.natDegree_add_le` is a max-bound, `natDegree_C_mul_le` kills the + scalar). This is the invariant behind `hdeg_out`: the committed output handle is a $G_1$ table + entry. + +* `degree_invariant` — $B = D \cdot 2^{\#\mathrm{mul}}$ for the flat table with products. A + product's degree is bounded by the sum of its operands' (`Polynomial.natDegree_mul_le`), so each + `mul` over a flat table can at worst double the running bound; the uniform flat bound is + exponential in the product count, not $2 D$. + +* `flat_2D_bound_false` — the naive "$B = 2 D$ once products are allowed" claim is false for the + flat table, proved by nesting one product inside another (`[seed, mul, mul]` builds $X^4$ at + $D = 1$). + +* `degree_invariant_paired` — $B = 2 D$ is recovered in a separate pairing-disciplined peer model: + a two-sorted table ($G_1$ / $G_t$) where products land in $G_t$ and hence never nest. + `GgmAdaptive.Move` itself has no pairing constructor, so this peer is not an invariant for its + operational run and is not used by the end-to-end theorem. + +## References + +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +-/ + +open Polynomial + +namespace GgmDegreeInvariant + +variable {p : ℕ} + +/-! ## The table operations and their interpreter -/ + +/-- A peer-model table-extension move: seeding with SRS powers, appending a linear combination, +or appending a formal product. Only the linear case mirrors `GgmAdaptive.runAux`; its actual +`Move` type has no product/pairing constructor. -/ +inductive TableOp (p : ℕ) where + /-- Append the SRS seed `1, X, …, X^D`. -/ + | seed : TableOp p + /-- Append `a·table[i] + b·table[j]` (group op + scalar mul, as `GgmAdaptive.combine`). -/ + | linComb (i j : ℕ) (a b : ZMod p) : TableOp p + /-- Append `table[i] * table[j]` (pairing product). -/ + | mul (i j : ℕ) : TableOp p + +/-- The SRS seed table `[1, X, …, X^D]`. -/ +noncomputable def srs (p D : ℕ) : List ((ZMod p)[X]) := + (List.range (D + 1)).map (fun k => X ^ k) + +/-- One table-extension step. Out-of-range handles read the `0` polynomial, exactly as +`GgmAdaptive` does everywhere (`List.getD _ _ 0`). -/ +noncomputable def applyOp (D : ℕ) (table : List ((ZMod p)[X])) : + TableOp p → List ((ZMod p)[X]) + | .seed => table ++ srs p D + | .linComb i j a b => table ++ [C a * table.getD i 0 + C b * table.getD j 0] + | .mul i j => table ++ [table.getD i 0 * table.getD j 0] + +/-- Build the handle table from an op list (head = last op applied), starting empty. -/ +noncomputable def buildTable (D : ℕ) : List (TableOp p) → List ((ZMod p)[X]) + | [] => [] + | op :: ops => applyOp D (buildTable D ops) op + +/-- The number of product ops in an op list — the doubling count of the flat bound. -/ +def mulCount : List (TableOp p) → ℕ + | [] => 0 + | .mul _ _ :: ops => mulCount ops + 1 + | .seed :: ops => mulCount ops + | .linComb _ _ _ _ :: ops => mulCount ops + +/-! ## Degree bookkeeping helpers -/ + +/-- A defaulted table read inherits any degree bound on the table (`0` has `natDegree 0`). -/ +lemma natDegree_getD_le {table : List ((ZMod p)[X])} {B : ℕ} + (h : ∀ q ∈ table, q.natDegree ≤ B) (i : ℕ) : + (table.getD i 0).natDegree ≤ B := by + by_cases hi : i < table.length + · rw [List.getD_eq_getElem _ _ hi] + exact h _ (List.getElem_mem hi) + · rw [List.getD_eq_default _ _ (Nat.le_of_not_lt hi)] + simp + +/-- A binary linear combination of two bounded reads stays bounded: `natDegree_add_le` is a +MAX-bound and `natDegree_C_mul_le` absorbs the scalars: the linComb step preserves ANY bound. -/ +lemma natDegree_linEntry_le {table : List ((ZMod p)[X])} {B : ℕ} + (h : ∀ q ∈ table, q.natDegree ≤ B) (i j : ℕ) (a b : ZMod p) : + (C a * table.getD i 0 + C b * table.getD j 0).natDegree ≤ B := by + refine (natDegree_add_le _ _).trans (max_le ?_ ?_) + · exact (natDegree_C_mul_le a _).trans (natDegree_getD_le h i) + · exact (natDegree_C_mul_le b _).trans (natDegree_getD_le h j) + +/-- Every seed polynomial `X^k`, `k ≤ D`, has `natDegree ≤ D` (`natDegree_X_pow_le`; the +`≤` direction needs no nontriviality). -/ +lemma natDegree_srs_le (D : ℕ) : ∀ q ∈ srs p D, q.natDegree ≤ D := by + intro q hq + obtain ⟨k, hk, rfl⟩ := List.mem_map.mp hq + exact (natDegree_X_pow_le k).trans (Nat.lt_succ_iff.mp (List.mem_range.mp hk)) + +/-! ## The flat-table invariants -/ + +/-- **The linComb-only degree invariant, B = D.** If the op list contains no product, every +table polynomial has `natDegree ≤ D` — by induction on the op list: the seed meets the bound +exactly and a linear combination degrades to the max of its operands. This is the structural +fact behind `GgmAdaptive`'s `hdeg_out` hypothesis (the committed output is a G₁ handle). -/ +theorem degree_invariant_linComb (D : ℕ) (ops : List (TableOp p)) + (hops : mulCount ops = 0) : + ∀ q ∈ buildTable D ops, q.natDegree ≤ D := by + induction ops with + | nil => intro q hq; simp [buildTable] at hq + | cons op ops ih => + intro q hq + cases op with + | seed => + simp only [buildTable, applyOp] at hq + rcases List.mem_append.mp hq with h | h + · exact ih hops q h + · exact natDegree_srs_le D q h + | linComb i j a b => + simp only [buildTable, applyOp] at hq + rcases List.mem_append.mp hq with h | h + · exact ih hops q h + · rw [List.mem_singleton.mp h] + exact natDegree_linEntry_le (ih hops) i j a b + | mul i j => simp [mulCount] at hops + +/-- **The flat-table degree invariant with products: B = D · 2^(#mul).** By induction on the +op list: seed entries have degree ≤ D ≤ B, a linear combination keeps the running bound +(max), and a product at worst DOUBLES it (`natDegree_mul_le` is a SUM-bound), consuming one +`mul` from the count. Over a FLAT table this exponential bound is the honest one — a uniform +`2·D` is refuted below (`flat_2D_bound_false`); recovering `2·D` needs the pairing +discipline (`degree_invariant_paired`). -/ +theorem degree_invariant (D : ℕ) (ops : List (TableOp p)) : + ∀ q ∈ buildTable D ops, q.natDegree ≤ D * 2 ^ mulCount ops := by + induction ops with + | nil => intro q hq; simp [buildTable] at hq + | cons op ops ih => + intro q hq + have hDB : D ≤ D * 2 ^ mulCount ops := + le_mul_of_one_le_right (Nat.zero_le D) Nat.one_le_two_pow + cases op with + | seed => + simp only [buildTable, applyOp] at hq + rcases List.mem_append.mp hq with h | h + · exact ih q h + · exact (natDegree_srs_le D q h).trans hDB + | linComb i j a b => + simp only [buildTable, applyOp] at hq + rcases List.mem_append.mp hq with h | h + · exact ih q h + · rw [List.mem_singleton.mp h] + exact natDegree_linEntry_le ih i j a b + | mul i j => + simp only [buildTable, applyOp] at hq + have hstep : D * 2 ^ mulCount ops + D * 2 ^ mulCount ops + = D * 2 ^ mulCount (.mul i j :: ops) := by + show _ = D * 2 ^ (mulCount ops + 1) + rw [pow_succ]; ring + rcases List.mem_append.mp hq with h | h + · exact (ih q h).trans (by rw [← hstep]; exact Nat.le_add_right _ _) + · rw [List.mem_singleton.mp h] + refine natDegree_mul_le.trans ?_ + rw [← hstep] + exact Nat.add_le_add (natDegree_getD_le ih i) (natDegree_getD_le ih j) + +/-- With at most ONE product op the flat bound specializes to `2·D` — the single honest +`2·D` statement available on the flat table. -/ +theorem degree_invariant_one_mul (D : ℕ) (ops : List (TableOp p)) + (hops : mulCount ops ≤ 1) : + ∀ q ∈ buildTable D ops, q.natDegree ≤ 2 * D := by + intro q hq + refine (degree_invariant D ops q hq).trans ?_ + calc D * 2 ^ mulCount ops + ≤ D * 2 ^ 1 := Nat.mul_le_mul_left D (Nat.pow_le_pow_right (by norm_num) hops) + _ = 2 * D := by ring + +/-- **The naive uniform `2·D` bound is FALSE on the flat table** (why `degree_invariant` +carries `2^(#mul)`): nesting one product inside another — `[seed, mul 1 1, mul 2 2]`, applied +seed-first — builds `X·X` and then `(X·X)·(X·X) = X^4` at `D = 1`, and `4 > 2·1`. Products +over a flat table COMPOUND; only the pairing discipline (below) forbids that. -/ +theorem flat_2D_bound_false [Fact (Nat.Prime p)] : + ∃ (D : ℕ) (ops : List (TableOp p)), + ¬ ∀ q ∈ buildTable D ops, q.natDegree ≤ 2 * D := by + refine ⟨1, [.mul 2 2, .mul 1 1, .seed], fun hall => ?_⟩ + have hsrs : srs p 1 = [X ^ 0, X ^ 1] := by + simp [srs, List.range_succ] + have htab : buildTable (p := p) 1 [.mul 2 2, .mul 1 1, .seed] + = [X ^ 0, X ^ 1, X ^ 1 * X ^ 1, X ^ 1 * X ^ 1 * (X ^ 1 * X ^ 1)] := by + simp only [buildTable, applyOp, hsrs, List.nil_append] + norm_num [List.getD] + have hmem : (X ^ 1 * X ^ 1 * (X ^ 1 * X ^ 1) : (ZMod p)[X]) + ∈ buildTable (p := p) 1 [.mul 2 2, .mul 1 1, .seed] := by + rw [htab]; simp + have hle := hall _ hmem + have h4 : (X ^ 1 * X ^ 1 * (X ^ 1 * X ^ 1) : (ZMod p)[X]).natDegree = 4 := by + have hX : (X ^ 1 * X ^ 1 * (X ^ 1 * X ^ 1) : (ZMod p)[X]) = X ^ 4 := by ring + rw [hX, natDegree_X_pow] + omega + +/-! ## The pairing-disciplined invariant: B = 2·D, structurally + +In this separate peer model, a product lands in Gₜ, out of which no further product is available. +Products therefore never nest. The two-sorted table below makes that discipline structural, and +the uniform bound `2·D` becomes an induction invariant. This section is not wired to +`GgmAdaptive.runAux`. -/ + +/-- A pairing-disciplined move: linear combinations within each sort, and a pairing product +whose operands are BOTH drawn from the G₁ table (degree ≤ D each — the faithful G₁ × G₂ +version only lowers the Gₜ bound to D + 1) and whose result is appended to the Gₜ table. -/ +inductive PairedOp (p : ℕ) where + /-- Append `a·g1[i] + b·g1[j]` to the G₁ table. -/ + | linG1 (i j : ℕ) (a b : ZMod p) : PairedOp p + /-- Append the pairing product `g1[i] * g1[j]` to the Gₜ table. -/ + | pair (i j : ℕ) : PairedOp p + /-- Append `a·gt[i] + b·gt[j]` to the Gₜ table. -/ + | linGt (i j : ℕ) (a b : ZMod p) : PairedOp p + +/-- Build the two-sorted `(G₁, Gₜ)` tables, seeding G₁ with the SRS. -/ +noncomputable def buildPaired (D : ℕ) : + List (PairedOp p) → List ((ZMod p)[X]) × List ((ZMod p)[X]) + | [] => (srs p D, []) + | .linG1 i j a b :: ops => + ((buildPaired D ops).1 + ++ [C a * (buildPaired D ops).1.getD i 0 + C b * (buildPaired D ops).1.getD j 0], + (buildPaired D ops).2) + | .pair i j :: ops => + ((buildPaired D ops).1, + (buildPaired D ops).2 ++ [(buildPaired D ops).1.getD i 0 * (buildPaired D ops).1.getD j 0]) + | .linGt i j a b :: ops => + ((buildPaired D ops).1, + (buildPaired D ops).2 + ++ [C a * (buildPaired D ops).2.getD i 0 + C b * (buildPaired D ops).2.getD j 0]) + +/-- **The pairing-disciplined degree invariant.** By induction on the op list, jointly: +every G₁ entry has `natDegree ≤ D` (seed + max under linear combination) and every Gₜ entry +has `natDegree ≤ 2·D` (a pairing product SUMS two G₁ bounds — `natDegree_mul_le` — and Gₜ +linear combinations keep the max). No hypotheses: the bound the `GgmAdaptive` theorems +consume as `hdeg_out` / `hdeg_pairs` (via differences, `Δ = 2·D`) is structural here. -/ +theorem degree_invariant_paired (D : ℕ) (ops : List (PairedOp p)) : + (∀ q ∈ (buildPaired D ops).1, q.natDegree ≤ D) ∧ + (∀ q ∈ (buildPaired D ops).2, q.natDegree ≤ 2 * D) := by + induction ops with + | nil => + refine ⟨natDegree_srs_le D, ?_⟩ + intro q hq + simp [buildPaired] at hq + | cons op ops ih => + obtain ⟨ih1, ih2⟩ := ih + cases op with + | linG1 i j a b => + refine ⟨?_, ih2⟩ + intro q hq + simp only [buildPaired] at hq + rcases List.mem_append.mp hq with h | h + · exact ih1 q h + · rw [List.mem_singleton.mp h] + exact natDegree_linEntry_le ih1 i j a b + | pair i j => + refine ⟨ih1, ?_⟩ + intro q hq + simp only [buildPaired] at hq + rcases List.mem_append.mp hq with h | h + · exact ih2 q h + · rw [List.mem_singleton.mp h] + refine natDegree_mul_le.trans ?_ + have h1 := natDegree_getD_le ih1 i + have h2 := natDegree_getD_le ih1 j + omega + | linGt i j a b => + refine ⟨ih1, ?_⟩ + intro q hq + simp only [buildPaired] at hq + rcases List.mem_append.mp hq with h | h + · exact ih2 q h + · rw [List.mem_singleton.mp h] + exact natDegree_linEntry_le ih2 i j a b + +/-- **Uniform B = 2·D under the pairing discipline** — every handle of either sort. -/ +theorem degree_invariant_paired_uniform (D : ℕ) (ops : List (PairedOp p)) : + ∀ q ∈ (buildPaired D ops).1 ++ (buildPaired D ops).2, q.natDegree ≤ 2 * D := by + intro q hq + rcases List.mem_append.mp hq with h | h + · exact ((degree_invariant_paired D ops).1 q h).trans (by omega) + · exact (degree_invariant_paired D ops).2 q h + +end GgmDegreeInvariant diff --git a/ArkLib/Scratch/KzgVacuity/GgmEmbed.lean b/ArkLib/Scratch/KzgVacuity/GgmEmbed.lean new file mode 100644 index 0000000000..e0ff8041cf --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmEmbed.lean @@ -0,0 +1,418 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Scratch.KzgVacuity.GgmArkLibTransport +import ArkLib.Scratch.KzgVacuity.GgmRandomEncoding +import ArkLib.Commitments.Functional.KZG.HardnessAssumptions + +/-! +# Embedding generic strategies into ArkLib's t-SDH adversary + +This file connects the generic-oracle strategy model (`GgmAdaptive.Strat` / `runAux`) to the +`Groups.tSdhAdversary` / `tSdhExperiment` interface, providing the construction that lets an +end-to-end t-SDH generic-group theorem quantify over a meaningful adversary class. Quantifying +over *all* `tSdhAdversary` is vacuous: the bound is false because a `Classical.choice`-definable +adversary computes discrete logarithms and wins with probability $1$. The remedy is to quantify +over generic **strategies** and apply `embed`; "generic-restricted" means *in the image of +`embed`*. The generic group model is due to Shoup [Sho97], with an explicit-equality abstract +model due to Maurer [Mau05]; the $t$-SDH assumption is that of Boneh and Boyen [BB04]. + +## The construction + +`embed strat : tSdhAdversary D` is `fun srs => pure (runEmbed g₁ D fuel strat srs)`. `runEmbed` +receives only the SRS group vectors (never $\tau$), seeds a `List G₁` handle table from the $G_1$ +tower, interprets `strat`'s linear-combination moves as real group products (`combineG`), answers +`strat`'s equality queries by real group equality (`DecidableEq`, classically) of the realized +handles, and returns the committed `(offset, G₁ elt)`. It never inverts the encoding: the opacity +is discharged by construction, since `strat : List Bool → …` receives only equality booleans. This +is why `embed strat` stays within the generic bound — it can only produce $g_1^{f(\tau)}$ with +$\deg f \le D$. `embed_run_correspondence` certifies that `runEmbed` reproduces the symbolic run's +output realized in the group, so the adversary is generic within this deterministic, one-sort, +explicit-equality model (a $\tau$-inverting cheat would break the correspondence). This is an +embedded subclass, not a characterization of all randomized or two-sort generic-group adversaries; +in particular `runEmbed` ignores the $G_2$ leg. + +## The crux + +In `embed_run_correspondence`, `runEmbed`'s equality branch compares real group elements +$g_1^{f(\tau)} \overset{?}{=} g_1^{h(\tau)}$; by injectivity (`gpow_val_inj_iff`) this equals +$f(\tau) \overset{?}{=} h(\tau)$, which is `GgmAdaptive.realAns τ f h`. So the group-table run +steps in lockstep with `runAux (realAns τ)`, threading the table-to-polynomial invariant +$\mathrm{tableG}[i] = g_1^{(\mathrm{table}[i])(\tau)}$ by induction on fuel (mirroring `runAux`'s +recursion). The invariant is seeded because the $G_1$ tower `PowerSrs.generate D τ` realizes each +seed monomial $X^k$ as $g_1^{\tau^k}$. + +## Scope + +The seed carries `1 ≤ D`: `srsSt D`'s table $[1, X, \ldots, X^D, 1, X]$ includes the two "$G_2$ +seed" monomials $1 = X^0$ and $X = X^1$, which the pairing-free $G_1$ adversary must realize from +its own $G_1$ tower $[g_1^{\tau^0}, \ldots, g_1^{\tau^D}]$. The handle $g_1^\tau = g_1^{\tau^1}$ is +present exactly when $D \ge 1$. At $D = 0$ the SRS is $(g_1), (g_2, g_2^\tau)$ and a $G_1$-output +adversary genuinely cannot form $g_1^\tau$ (there is no pairing to move $g_2^\tau$ into $G_1$), so +the unconditional correspondence is false there — an honest fact about the interface, not a proof +gap. The regime $D \ge 1$ is the meaningful KZG one. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +* [Maurer, U., *Abstract Models of Computation in Cryptography*][Mau05] +-/ + +open Polynomial Groups + +namespace GgmEmbed + +open GgmCandidate GgmAdaptive GgmRandomEncoding GgmArkLibTransport + +variable {p : ℕ} [Fact (Nat.Prime p)] +variable {G₁ : Type} [Group G₁] [PrimeOrderWith G₁ p] +variable {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] + +/-! ## 1. The exponent encoding `E g a = g ^ a.val` as an additive-to-multiplicative morphism. + +`E g` sends the field of exponents into the group; the correspondence realizes every handle through +it. Its two morphism laws (over an order-`p` base) are all we need to push `combine`'s field-linear +combination through to `combineG`'s group-linear combination. -/ + +lemma encode_zero (g : G₁) : g ^ (0 : ZMod p).val = 1 := by simp + +omit [PrimeOrderWith G₁ p] in +/-- `E g (a + b) = E g a * E g b`. -/ +lemma encode_add {g : G₁} (hord : orderOf g = p) (a b : ZMod p) : + g ^ (a + b).val = g ^ a.val * g ^ b.val := by + rw [← pow_add] + exact (Groups.gpow_eq_of_nat_cast_eq hord _ _ (by push_cast [ZMod.natCast_zmod_val]; ring)).symm + +omit [PrimeOrderWith G₁ p] in +/-- `E g (c * a) = (E g a) ^ c.val` — scalar multiplication becomes group exponentiation. -/ +lemma encode_mul {g : G₁} (hord : orderOf g = p) (c a : ZMod p) : + g ^ (c * a).val = (g ^ a.val) ^ c.val := by + rw [← pow_mul] + exact (Groups.gpow_eq_of_nat_cast_eq hord _ _ (by push_cast [ZMod.natCast_zmod_val]; ring)).symm + +/-! ## 2. The group-side linear-combination oracle and the table↔polynomial invariant. -/ + +/-- The real-group realization of a `Move.lin` move: `∏ᵢ (tableG[idxᵢ]) ^ (cᵢ).val` — the group +product-of-powers that faithfully realizes `GgmAdaptive.combine`'s formal `Σᵢ cᵢ · table[idxᵢ]`. -/ +noncomputable def combineG (spec : List (ZMod p × ℕ)) (tableG : List G₁) : G₁ := + (spec.map (fun ci => (tableG.getD ci.2 1) ^ ci.1.val)).prod + +/-- The table↔polynomial invariant threaded through the run: the group table and the symbolic +polynomial table have equal length and each group handle is the encoding of its polynomial's +evaluation at `τ`. (The pointwise clause holds even off the ends: both default to +`1 = g^(0.eval τ)`.) -/ +def IsEncoding (g : G₁) (τ : ZMod p) (tableG : List G₁) (table : List ((ZMod p)[X])) : Prop := + tableG.length = table.length ∧ + ∀ i, tableG.getD i 1 = g ^ ((table.getD i 0).eval τ).val + +/-- **The combine correspondence.** Under the invariant, the group product-of-powers realizes the +encoding of the formal linear combination's evaluation. Induction on the move's index list. -/ +lemma combineG_eq {g : G₁} (hord : orderOf g = p) (τ : ZMod p) + (spec : List (ZMod p × ℕ)) (tableG : List G₁) (table : List ((ZMod p)[X])) + (hInv : IsEncoding g τ tableG table) : + combineG spec tableG = g ^ ((combine spec table).eval τ).val := by + induction spec with + | nil => simp [combineG, combine] + | cons ck t ih => + have hcomb : combine (ck :: t) table + = Polynomial.C ck.1 * table.getD ck.2 0 + combine t table := by + simp [combine, List.map_cons, List.sum_cons] + rw [combineG, List.map_cons, List.prod_cons, ← combineG, ih, hcomb] + rw [eval_add, eval_mul, eval_C, encode_add hord, encode_mul hord, hInv.2 ck.2] + +/-! ## 3. `runEmbed` — the real-group run, and `embed` — the ArkLib adversary. -/ + +/-- REAL group equality as a `Bool`, decided classically (an abstract group carries no decision +procedure). This is the query oracle: `strat` learns only this boolean, never a group element. -/ +noncomputable def groupEq (x y : G₁) : Bool := @decide (x = y) (Classical.propDecidable _) + +/-- **The real-group generic run.** A `List G₁` handle table (no polynomials, no τ), evolved by +`strat`'s moves interpreted as REAL group operations: `Move.lin` appends a `combineG` product, +`Move.query i j` appends the boolean of REAL group equality `tableG[i] = tableG[j]`, the committed +output reads the offset and the output handle out of the table. Mirrors `GgmAdaptive.runAux`'s +recursion exactly. -/ +noncomputable def runEmbedAux (g : G₁) (strat : Strat p) : + ℕ → (List G₁ × List Bool) → Option (ZMod p × G₁) + | 0, _ => some (0, 1) + | fuel + 1, (tableG, hist) => + match strat hist with + | Sum.inr (c, k) => some (c, tableG.getD k 1) + | Sum.inl (Move.lin spec) => + runEmbedAux g strat fuel (tableG ++ [combineG spec tableG], hist) + | Sum.inl (Move.query i j) => + runEmbedAux g strat fuel (tableG, hist ++ [groupEq (tableG.getD i 1) (tableG.getD j 1)]) + +/-- The G₁ seed table: the tower handles `g₁^(τ^k)` (`k ≤ D`) plus the two "G₂ seed" monomials +`1 = X^0` and `X = X^1` realized from the same tower (positions `0` and `1`) — matching `srsSt D`'s +polynomial table `[X^0,…,X^D, 1, X]` entry-for-entry. -/ +noncomputable def seedG (srs1 : List G₁) (D : ℕ) : List G₁ := + ((List.range (D + 1)).map (fun i => srs1.getD i 1)) ++ [srs1.getD 0 1, srs1.getD 1 1] + +/-- **`runEmbed`** — run `strat` against the real-group SRS. A function of the G₁ generator `g₁` +and the SRS alone: it reads only `srs.1` (the G₁ tower), so — being pairing-free — it needs neither +the G₂ generator, nor `srs.2`, nor τ. -/ +noncomputable def runEmbed (g₁ : G₁) (D fuel : ℕ) (strat : Strat p) + (srs : Vector G₁ (D + 1) × Vector G₂ 2) : Option (ZMod p × G₁) := + runEmbedAux g₁ strat fuel (seedG srs.1.toList D, []) + +/-- **`embed : Strat p → tSdhAdversary D`.** Deterministic, empty-cache; its IMAGE is the +"generic-restricted" adversary class the target theorem quantifies over. Pairing-free: a function of +`g₁` and the SRS only (the G₂ generator is never consulted). -/ +noncomputable def embed (g₁ : G₁) (D fuel : ℕ) (strat : Strat p) : + Groups.tSdhAdversary D (G₁ := G₁) (G₂ := G₂) (p := p) := + fun srs => pure (runEmbed g₁ D fuel strat srs) + +/-! ## 4. The correspondence: `runEmbedAux` steps in lockstep with `runAux (realAns τ)`. -/ + +/-- Appending a matching pair of handles preserves the invariant: the new group handle realizes the +new polynomial (`combineG_eq`), and existing handles are untouched. -/ +lemma isEncoding_append {g : G₁} (hord : orderOf g = p) {τ : ZMod p} + {tableG : List G₁} {table : List ((ZMod p)[X])} (spec : List (ZMod p × ℕ)) + (hInv : IsEncoding g τ tableG table) : + IsEncoding g τ (tableG ++ [combineG spec tableG]) (table ++ [combine spec table]) := by + obtain ⟨hlen, hpt⟩ := hInv + refine ⟨by simp [hlen], fun i => ?_⟩ + rcases lt_trichotomy i tableG.length with h | h | h + · rw [List.getD_append _ _ _ _ h, List.getD_append _ _ _ _ (hlen ▸ h)] + exact hpt i + · subst h + rw [List.getD_append_right _ _ _ _ (le_refl _), + List.getD_append_right _ _ _ _ (by rw [hlen]), hlen] + simp only [Nat.sub_self, List.getD_cons_zero] + exact combineG_eq hord τ spec tableG table ⟨hlen, hpt⟩ + · rw [List.getD_append_right _ _ _ _ (le_of_lt h), + List.getD_append_right _ _ _ _ (by rw [← hlen]; exact le_of_lt h)] + rw [List.getD_eq_default _ _ (by simpa using (by omega : 1 ≤ i - tableG.length)), + List.getD_eq_default _ _ (by rw [← hlen]; simpa using (by omega : 1 ≤ i - tableG.length))] + simp + +/-- **THE CORRESPONDENCE (induction core).** Under the invariant, `runEmbedAux` on the group table +returns exactly the committed offset of `runAux (realAns τ)` and the real-group encoding of its +committed output polynomial — for the SAME history. The two runs step in lockstep: equality queries +agree because `realAns τ` answers `f.eval τ =? h.eval τ`, which injectivity (`gpow_val_inj_iff`) +aligns with the real group equality `g^(f τ).val =? g^(h τ).val`. -/ +lemma runEmbedAux_correspondence {g : G₁} (hord : orderOf g = p) (τ : ZMod p) (strat : Strat p) : + ∀ (fuel : ℕ) (tableG : List G₁) (table : List ((ZMod p)[X])) (hist : List Bool), + IsEncoding g τ tableG table → + runEmbedAux g strat fuel (tableG, hist) + = some ((runAux (realAns τ) strat fuel ⟨table, hist⟩).1.1, + g ^ (((runAux (realAns τ) strat fuel ⟨table, hist⟩).1.2).eval τ).val) := by + intro fuel + induction fuel with + | zero => + intro tableG table hist hInv + simp [runEmbedAux, runAux] + | succ fuel ih => + intro tableG table hist hInv + rcases hdec : strat hist with m | out + · cases m with + | lin spec => + have e1 : runEmbedAux g strat (fuel + 1) (tableG, hist) + = runEmbedAux g strat fuel (tableG ++ [combineG spec tableG], hist) := by + simp only [runEmbedAux, hdec] + have e2 : runAux (realAns τ) strat (fuel + 1) ⟨table, hist⟩ + = runAux (realAns τ) strat fuel ⟨table ++ [combine spec table], hist⟩ := by + simp only [runAux, hdec] + rw [e1, e2] + exact ih _ _ _ (isEncoding_append hord spec hInv) + | query i j => + have hans : groupEq (tableG.getD i 1) (tableG.getD j 1) + = realAns τ (table.getD i 0) (table.getD j 0) := by + simp only [groupEq, realAns] + rw [decide_eq_decide, hInv.2 i, hInv.2 j] + exact gpow_val_inj_iff hord + have e1 : runEmbedAux g strat (fuel + 1) (tableG, hist) + = runEmbedAux g strat fuel + (tableG, hist ++ [groupEq (tableG.getD i 1) (tableG.getD j 1)]) := by + simp only [runEmbedAux, hdec] + have e2 : runAux (realAns τ) strat (fuel + 1) ⟨table, hist⟩ + = ((runAux (realAns τ) strat fuel + ⟨table, hist ++ [realAns τ (table.getD i 0) (table.getD j 0)]⟩).1, + (table.getD i 0, table.getD j 0) :: + (runAux (realAns τ) strat fuel + ⟨table, hist ++ [realAns τ (table.getD i 0) (table.getD j 0)]⟩).2) := by + simp only [runAux, hdec] + rw [e1, e2, hans] + exact ih _ _ _ hInv + · have e1 : runEmbedAux g strat (fuel + 1) (tableG, hist) + = some (out.1, tableG.getD out.2 1) := by + simp only [runEmbedAux, hdec] + have e2 : runAux (realAns τ) strat (fuel + 1) ⟨table, hist⟩ + = ((out.1, table.getD out.2 0), []) := by + simp only [runAux, hdec] + rw [e1, e2, hInv.2 out.2] + +/-! ## 5. The seed invariant and the deliverable correspondence. -/ + +/-- `getD` of a `List.range`-map at an in-range index. -/ +lemma rangeMap_getD_lt {α : Type*} (f : ℕ → α) (n i : ℕ) (d : α) (h : i < n) : + ((List.range n).map f).getD i d = f i := by + have hlen : i < ((List.range n).map f).length := by simpa using h + rw [List.getD_eq_getElem?_getD, List.getElem?_eq_getElem hlen, Option.getD_some, + List.getElem_map, List.getElem_range] + +/-- Tower `toList` indexing: the `i`-th tower handle (`i ≤ D`) is `g₁^(τ^i)`. -/ +lemma tower_toList_getD {g : G₁} (τ : ZMod p) (D i : ℕ) (h : i < D + 1) : + (PowerSrs.tower g τ D).toList.getD i 1 = g ^ (τ.val ^ i) := by + unfold PowerSrs.tower + rw [Vector.toList_ofFn, List.getD_eq_getElem?_getD, List.getElem?_ofFn] + rw [dif_pos h] + rfl + +/-- **The seed invariant.** For `srs = generate D τ` with `1 ≤ D`, the G₁ seed table realizes +`srsSt D`'s polynomial table under the encoding: the tower positions cover `X^0,…,X^D`, and the two +"G₂ seed" monomials `1, X` are `g₁^(τ^0), g₁^(τ^1)`. -/ +lemma seedG_isEncoding (g : G₁) (hord : orderOf g = p) (τ : ZMod p) (D : ℕ) (hD : 1 ≤ D) : + IsEncoding g τ (seedG (PowerSrs.tower g τ D).toList D) (srsSt D).table := by + have htab : (srsSt (p := p) D).table + = ((List.range (D + 1)).map (fun i => (X : (ZMod p)[X]) ^ i)) ++ [1, X] := rfl + refine ⟨?_, fun i => ?_⟩ + · rw [seedG, htab]; simp + · -- realize each seed monomial's evaluation + have key : ∀ k, k < D + 1 → + (PowerSrs.tower g τ D).toList.getD k 1 = g ^ ((X ^ k : (ZMod p)[X]).eval τ).val := by + intro k hk + rw [tower_toList_getD τ D k hk, eval_pow, eval_X] + exact Groups.gpow_eq_of_nat_cast_eq hord _ _ (by push_cast [ZMod.natCast_zmod_val]; ring) + rw [seedG, htab] + rcases lt_trichotomy i (D + 1) with h | h | h + · -- first segment: X^i + rw [List.getD_append _ _ _ _ (by simpa using h), + List.getD_append _ _ _ _ (by simpa using h), + rangeMap_getD_lt _ _ _ _ h, rangeMap_getD_lt _ _ _ _ h] + exact key i h + · -- position D+1 : the "G₂ seed" 1 = X^0 + subst h + rw [List.getD_append_right _ _ _ _ (by simp), + List.getD_append_right _ _ _ _ (by simp)] + simp only [List.length_map, List.length_range, Nat.sub_self, List.getD_cons_zero, eval_one] + have := key 0 (Nat.succ_pos D) + rw [eval_pow, eval_X, pow_zero] at this + simpa using this + · -- positions ≥ D+2 : the "G₂ seed" X (at D+2, needs D ≥ 1) then defaults + rw [List.getD_append_right _ _ _ _ (by simp only [List.length_map, List.length_range]; omega), + List.getD_append_right _ _ _ _ (by simp only [List.length_map, List.length_range]; omega)] + simp only [List.length_map, List.length_range] + rcases Nat.lt_or_ge i (D + 3) with h2 | h2 + · -- i = D + 2 : the "G₂ seed" X = X^1 + have hi : i - (D + 1) = 1 := by omega + rw [hi] + simp only [List.getD_cons_succ, List.getD_cons_zero, eval_X] + have := key 1 (by omega) + rw [eval_pow, eval_X, pow_one] at this + exact this + · -- i ≥ D + 3 : both default + rw [List.getD_eq_default _ _ (by simp only [List.length_cons, List.length_nil]; omega), + List.getD_eq_default _ _ (by simp only [List.length_cons, List.length_nil]; omega)] + simp + +omit [PrimeOrderWith G₂ p] in +/-- **THE DELIVERABLE (§2b, verbatim).** `runEmbed` on the real SRS `PowerSrs.generate D τ` returns +the committed offset of the SYMBOLIC run `runOutput (realAns τ) strat fuel (srsSt D)` and the +real-group encoding `g₁ ^ (output.eval τ).val` of its committed output polynomial. This certifies +`embed strat` is genuinely generic (it reproduces the symbolic run realized in the group, never +inverting the encoding), and is the socket the end-to-end composition (task E) consumes. -/ +theorem embed_run_correspondence {g₁ : G₁} {g₂ : G₂} (hord : orderOf g₁ = p) + (D : ℕ) (hD : 1 ≤ D) (τ : ZMod p) (strat : Strat p) (fuel : ℕ) : + runEmbed g₁ D fuel strat (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ) + = some ((runOutput (realAns τ) strat fuel (srsSt D)).1, + g₁ ^ ((runOutput (realAns τ) strat fuel (srsSt D)).2.eval τ).val) := by + have hsrs1 : (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ).1 = PowerSrs.tower g₁ τ D := rfl + rw [runEmbed, hsrs1, + runEmbedAux_correspondence hord τ strat fuel + (seedG (PowerSrs.tower g₁ τ D).toList D) (srsSt (p := p) D).table [] + (seedG_isEncoding g₁ hord τ D hD)] + rfl + +/-! ## 6. Non-collapse: the embedded adversary class is a genuine non-singleton. + +`tSdh_ggm_sound` (GgmEndToEnd) quantifies over the IMAGE of `embed`, NOT over all +`Groups.tSdhAdversary` (over which the bound is FALSE — a `Classical.choice`-definable adversary +inverts the encoding and wins with probability 1). A reviewer must therefore see that the +replacement class is at least not degenerate: `embed_noncollapsing` certifies that its image is a +genuine non-singleton and really exercises the group. This richness witness does not prove that the +image captures every conventional generic-group adversary. + +⚑ WHAT THIS WITNESSES (and what it does NOT). It witnesses only RICHNESS — the image of `embed` is +not a single point. It does NOT claim completeness of the model or injectivity of `embed`: literal +injectivity is FALSE, and honestly so. +Two strategies that differ only on equality-histories the run never actually reaches produce the +SAME committed adversary — `runEmbed` consults a strategy only along the single branch its own +realized queries carve out, so off-branch disagreement is invisible. The theorem therefore records +that the intended restricted class has distinct behaviors, without claiming an injective +parametrization. + +⚑ WHERE `g₁ ≠ 1` LIVES. This hypothesis belongs HERE, not on `tSdh_ggm_sound`. Soundness holds for +ANY base `g₁` — the counting bound never needs a nontrivial generator, and `tSdh_ggm_sound` +correctly does not assume one. But the *meaningfulness* of quantifying over `embed`'s image — that +the class carries a genuinely nontrivial GROUP output (conjunct (ii)), not merely distinct offset +labels — is what consumes `g₁ ≠ 1`. That asymmetry (soundness free of it, meaningfulness using it) +is itself the point. The non-singleton conjunct (i) needs even less: only `0 ≠ 1` in `ZMod p`. -/ + +/-- The constant strategy that immediately commits offset `c` and reads out handle `0`, ignoring the +equality history. The two instances `stratOffset 0` / `stratOffset 1` are the non-collapse +witness. -/ +private def stratOffset (c : ZMod p) : Strat p := fun _ => Sum.inr (c, 0) + +/-- `getD 0` of the seed table is the `0`-th SRS G₁ handle (the seed's first segment starts at the +tower, so position `0` is `srs1[0]`). -/ +lemma seedG_getD_zero (srs1 : List G₁) (D : ℕ) : + (seedG srs1 D).getD 0 1 = srs1.getD 0 1 := by + rw [seedG, List.getD_append _ _ _ _ (by simp), + rangeMap_getD_lt (fun i => srs1.getD i 1) (D + 1) 0 1 (Nat.succ_pos D)] + +omit [Group G₂] [PrimeOrderWith G₂ p] in +/-- A `stratOffset c` run (with at least one unit of fuel) commits `c` and the `0`-th seed handle — +which is the SRS's `0`-th G₁ element — on ANY SRS, decided with no group machinery. -/ +lemma runEmbed_stratOffset {g₁ : G₁} (D f : ℕ) (c : ZMod p) + (srs : Vector G₁ (D + 1) × Vector G₂ 2) : + runEmbed g₁ D (f + 1) (stratOffset c) srs = some (c, srs.1.toList.getD 0 1) := by + simp only [runEmbed, runEmbedAux, stratOffset, seedG_getD_zero] + +omit [PrimeOrderWith G₂ p] in +/-- **`embed_noncollapsing` — the non-collapse witness.** There are two generic strategies whose +`embed`-outputs (the deterministic `runEmbed` values, since `embed strat srs = pure (runEmbed …)`) +are distinct, so the IMAGE of `embed` is a genuine non-singleton: + +* (i) On EVERY SRS, `stratOffset 0` and `stratOffset 1` commit distinct offsets (`0 ≠ 1` in + `ZMod p`), hence distinct outputs — no group machinery, no nontrivial generator. This is the + non-vacuity of `tSdh_ggm_sound`'s quantifier: its range is not one point. +* (ii) On the real KZG SRS `PowerSrs.generate D τ`, `stratOffset 0`'s committed GROUP element is the + base generator `g₁`, which is `≠ 1` by `hg₁`. So the image really exercises the group — its + adversaries produce nontrivial group outputs, not only distinct labels. This is the sole consumer + of `g₁ ≠ 1`. + +Honest non-claim: this does not assert `embed` is injective (it is not — see the section note); it +asserts the image is not a singleton, which is what a meaningful quantifier requires. -/ +theorem embed_noncollapsing {g₁ : G₁} {g₂ : G₂} (hg₁ : g₁ ≠ 1) (D f : ℕ) : + ∃ s₀ s₁ : Strat p, + (∀ srs : Vector G₁ (D + 1) × Vector G₂ 2, + runEmbed g₁ D (f + 1) s₀ srs ≠ runEmbed g₁ D (f + 1) s₁ srs) ∧ + (∀ τ : ZMod p, + runEmbed g₁ D (f + 1) s₀ (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ) + ≠ some (0, 1)) := by + have h01 : (0 : ZMod p) ≠ 1 := zero_ne_one + refine ⟨stratOffset 0, stratOffset 1, ?_, ?_⟩ + · -- (i) distinct offsets on every SRS, by `0 ≠ 1` + intro srs h + rw [runEmbed_stratOffset D f 0 srs, runEmbed_stratOffset D f 1 srs, + Option.some.injEq, Prod.mk.injEq] at h + exact h01 h.1 + · -- (ii) the group output on the real SRS is the nontrivial generator `g₁` + intro τ h + have hval : (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ).1.toList.getD 0 1 = g₁ := by + have hsrs1 : (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ).1 = PowerSrs.tower g₁ τ D := rfl + rw [hsrs1, tower_toList_getD τ D 0 (Nat.succ_pos D), pow_zero, pow_one] + rw [runEmbed_stratOffset D f 0 (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ), hval, + Option.some.injEq, Prod.mk.injEq] at h + exact hg₁ h.2 + +#print axioms embed_run_correspondence +#print axioms embed +#print axioms runEmbed +#print axioms embed_noncollapsing + +end GgmEmbed diff --git a/ArkLib/Scratch/KzgVacuity/GgmEndToEnd.lean b/ArkLib/Scratch/KzgVacuity/GgmEndToEnd.lean new file mode 100644 index 0000000000..b30867c46b --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmEndToEnd.lean @@ -0,0 +1,219 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Scratch.KzgVacuity.GgmEmbed +import ArkLib.Scratch.KzgVacuity.GgmProbThreading +import ArkLib.Scratch.KzgVacuity.GgmDegreeDischarge + +/-! +# End-to-end t-SDH soundness in the generic group model + +This file proves the single end-to-end t-SDH soundness theorem about ArkLib's real +`tSdhExperiment`, in the generic group model of Shoup [Sho97] and Maurer [Mau05]. The headline +theorem is `tSdh_ggm_sound`, together with its companion `tSdh_ggm_sound_lt_one`. + +## Escaping vacuity + +The statement `∀ A : tSdhAdversary D, tSdhExperiment D A ≤ ε` is *false* for small `ε`: a +`Classical.choice`-definable adversary inverts the encoding and wins the Boneh–Boyen t-SDH +game [BB04] with probability `1`. So the theorem does not quantify over all `tSdhAdversary`. +Instead it quantifies over generic *strategies* `strat : Strat p` and applies the embedding +`GgmEmbed.embed`; the generic-restricted adversary class is the *image* of `embed`. Here `Strat` +is deterministic and fuel-bounded. `embed strat` receives only equality booleans +(`strat : List Bool → …`) — no group element is ever in scope for it to invert, following +Maurer's explicit-equality generic group model [Mau05] — so it can only realize $g_1^{f(τ)}$ +with $\deg f ≤ D$, which is exactly what the counting bound bounds. + +## The composition chain + +For every generic strategy `strat : Strat p`, the success probability of `embed strat` collapses +to a finite count and is bounded in three steps, yielding +$\texttt{tSdhExperiment}\ D\ (\texttt{embed strat}) \le +(\binom{fuel + D + 4}{2} \cdot D + (D + 1)) / (p - 1)$: + +* `experiment_eq_count` turns ArkLib's `tSdhExperiment` into a `Finset` count over + $\mathrm{Fin}\,(p-1)$, its determinism hypothesis discharged by `embed_det` (which rests on + `embed_run_correspondence`); +* the count is bounded by `realWinSet.card` via the injection `winIndex_card_le` and the transport + `groupWinSet_eq_realWinSet`; +* `realWinSet.card` is bounded by `card_realWinSet_le_encoding_D` at $Δ = D$, with both degree + hypotheses discharged by the `_of_run` theorems about the real `runTable`. + +Every gluing lemma is about the real objects: the degree discharge concerns the actual `runTable`; +the count is `experiment_eq_count` about ArkLib's `tSdhExperiment`; the correspondence is +`embed_run_correspondence` about ArkLib's SRS. No peer model or restated definition is swapped into +any socket. + +## Side conditions + +The hypotheses are `1 ≤ D` (the meaningful KZG regime for this embedding; at `D = 0` the +pairing-free `G₁` adversary genuinely cannot form $g_1^τ$) and `orderOf g₁ = p` (the base is a +generator, used by encoding injectivity). Primality already implies `2 ≤ p`, so the headline +theorem does not take that redundant hypothesis. The `∀ i, SampleableType (unifSpec.Range i)` +instance is ArkLib's own assumption on `tSdhExperiment`, carried verbatim. `tSdh_ggm_sound_lt_one` +adds the standard regime hypothesis $\binom{fuel + D + 4}{2} \cdot D + (D + 1) < p - 1$ and +delivers a genuine `< 1`. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +* [Maurer, U., *Abstract Models of Computation in Cryptography*][Mau05] +-/ + +open Polynomial Groups OracleSpec OracleComp +open scoped Classical NNReal ENNReal + +namespace GgmEndToEnd + +open GgmCandidate GgmAdaptive GgmRandomEncoding GgmArkLibTransport GgmEmbed Ggm.ProbThreading +open GgmDegreeDischarge + +variable {p : ℕ} [Fact (Nat.Prime p)] + [∀ i, SampleableType (unifSpec.Range i)] + {G₁ : Type} [Group G₁] [PrimeOrderWith G₁ p] {g₁ : G₁} + {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] {g₂ : G₂} + {D : ℕ} + +/-! ## 1. The deterministic output of `embed strat`, and `embed`'s determinism. -/ + +/-- The (deterministic-given-τ) `Option`-output of `embed strat` on the SRS generated from `τ`: the +committed offset of the symbolic run realized in the group. This is the RHS of D's +`embed_run_correspondence`, packaged as the `resultOf` that C's `experiment_eq_count` consumes. -/ +noncomputable def stratResult (g₁ : G₁) (D fuel : ℕ) (strat : Strat p) : + ZMod p → Option (ZMod p × G₁) := + fun τ => some ((runOutput (realAns τ) strat fuel (srsSt D)).1, + g₁ ^ ((runOutput (realAns τ) strat fuel (srsSt D)).2.eval τ).val) + +omit [∀ i, SampleableType (unifSpec.Range i)] [PrimeOrderWith G₂ p] in +/-- **`embed strat` is deterministic-given-τ from the empty cache**, with output `stratResult`. +`embed strat srs = pure (runEmbed … srs)`, so `.run' ∅ = pure (runEmbed …)`, and D's +`embed_run_correspondence` identifies `runEmbed …` on `PowerSrs.generate D τ` with `stratResult τ`. +This is the exact `hdet` hypothesis of C's `game_collapse` / `experiment_eq_count`. -/ +theorem embed_det (hord₁ : orderOf g₁ = p) (hD : 1 ≤ D) (strat : Strat p) (fuel : ℕ) : + ∀ τ, (embed g₁ D fuel strat + (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ)).run' ∅ + = pure (stratResult g₁ D fuel strat τ) := by + intro τ + simp only [embed, StateT.run'_pure', stratResult] + rw [embed_run_correspondence hord₁ D hD τ strat fuel] + +/-! ## 2. The reindex `Fin (p−1) ↪ groupWinSet`: winning-index count ≤ winning-set card. -/ + +omit [∀ i, SampleableType (unifSpec.Range i)] in +/-- The nonzero-trapdoor index `i : Fin (p−1)` maps to a genuine nonzero residue `(i+1 : ZMod p)` +(matching `winPred`'s `((i : ℕ) + 1 : ZMod p)` — the ZMod-level `+`). -/ +lemma index_ne_zero (hp : 2 ≤ p) (i : Fin (p - 1)) : ((i : ℕ) + 1 : ZMod p) ≠ 0 := by + haveI : NeZero p := ⟨by omega⟩ + have hlt : (i : ℕ) + 1 < p := by have := i.isLt; omega + have hcast : ((i : ℕ) + 1 : ZMod p) = (((i : ℕ) + 1 : ℕ) : ZMod p) := by push_cast; ring + rw [hcast] + intro h + have h2 : (((i : ℕ) + 1 : ℕ) : ZMod p).val = 0 := by rw [h]; exact ZMod.val_zero + rw [ZMod.val_natCast_of_lt hlt] at h2 + omega + +omit [∀ i, SampleableType (unifSpec.Range i)] in +/-- **The reindex bound.** The number of winning nonzero-trapdoor indices `i : Fin (p−1)` (those on +which `embed strat`'s deterministic output `stratResult` wins ArkLib's `tSdhCondition`) is at most +the cardinality of the field-level winning set `realWinSet`. Proved by an injection +`i ↦ (i+1 : ZMod p)` into `realWinSet`, transporting each winning index through the condition +equivalence `tSdhCondition_iff_field` (the pointwise fact that `groupWinSet_eq_realWinSet` packages +as a set identity). Only the `≤` direction is needed, so no surjectivity is required. -/ +theorem winIndex_card_le (hord₁ : orderOf g₁ = p) (hp : 2 ≤ p) + (strat : Strat p) (fuel : ℕ) : + (Finset.univ.filter (winPred (stratResult g₁ D fuel strat) g₁)).card + ≤ (realWinSet strat (srsSt D) fuel).card := by + refine Finset.card_le_card_of_injOn (fun i => ((i : ℕ) + 1 : ZMod p)) ?_ ?_ + · -- maps winning indices into `realWinSet` + intro i hi + rw [Finset.mem_coe, Finset.mem_filter] at hi + have hw := hi.2 + simp only [winPred] at hw + obtain ⟨ch, hres, hcond⟩ := hw + -- `stratResult … (i+1) = some (offset, encoded output)` pins `ch` + rw [stratResult, Option.some.injEq] at hres + rw [← hres] at hcond + -- transport ArkLib's group condition to the field predicate `realWinSet` filters by + rw [Finset.mem_coe, realWinSet, Finset.mem_filter] + refine ⟨?_, (tSdhCondition_iff_field hord₁ _ _ _).mp hcond⟩ + rw [nonzeroPoints, Finset.mem_erase] + exact ⟨index_ne_zero hp i, Finset.mem_univ _⟩ + · -- `i ↦ (i+1 : ZMod p)` is injective on `Fin (p−1)` + intro i _ j _ hij + simp only at hij + have hi : (i : ℕ) + 1 < p := by have := i.isLt; omega + have hj : (j : ℕ) + 1 < p := by have := j.isLt; omega + have hci : ((i : ℕ) + 1 : ZMod p) = (((i : ℕ) + 1 : ℕ) : ZMod p) := by push_cast; ring + have hcj : ((j : ℕ) + 1 : ZMod p) = (((j : ℕ) + 1 : ℕ) : ZMod p) := by push_cast; ring + rw [hci, hcj] at hij + have := congrArg ZMod.val hij + rw [ZMod.val_natCast_of_lt hi, ZMod.val_natCast_of_lt hj] at this + exact Fin.ext (by omega) + +/-! ## 3. ⚑ THE CAPSTONE. -/ + +omit [PrimeOrderWith G₂ p] in +/-- **`tSdh_ggm_sound` — the single end-to-end t-SDH GGM soundness theorem.** For every generic +strategy `strat : Strat p`, the embedded ArkLib adversary `embed strat` wins ArkLib's REAL t-SDH + experiment with probability at most the conservative all-pairs collision number +`(C(fuel+D+4, 2)·D + (D+1)) / (p − 1)`. The whole argument is wired through ONE socket: + +* `experiment_eq_count` (C) turns ArkLib's `tSdhExperiment` into a `Finset` count over `Fin (p−1)`, + its determinism hypothesis discharged by `embed_det` (which rests on the embedding's + `embed_run_correspondence`); +* the count is bounded by `realWinSet.card` via the injection `winIndex_card_le` + the transport + `groupWinSet_eq_realWinSet`; +* `realWinSet.card` is bounded by `card_realWinSet_le_encoding_D` (A) at Δ = D, with BOTH degree + hypotheses discharged by B's `_of_run` theorems about the real `runTable`. + +The theorem is about the IMAGE of `embed`, a genuinely rich strategy space — NOT the full +`tSdhAdversary` type (over which the statement is false). -/ +theorem tSdh_ggm_sound + (hord₁ : orderOf g₁ = p) + (hD : 1 ≤ D) (strat : Strat p) (fuel : ℕ) : + tSdhExperiment (g₁ := g₁) (g₂ := g₂) D (embed g₁ D fuel strat) + ≤ (((fuel + D + 4).choose 2 * D + (D + 1) : ℕ) : ℝ≥0∞) / ((p - 1 : ℕ) : ℝ≥0∞) := by + have hp : 2 ≤ p := (Fact.out : Nat.Prime p).two_le + -- (C) collapse the experiment to a count over `Fin (p−1)` + rw [experiment_eq_count D (embed g₁ D fuel strat) (stratResult g₁ D fuel strat) + (embed_det hord₁ hD strat fuel)] + -- the numerator is bounded, in ℕ, by the Shoup number + have hcard : (Finset.univ.filter (winPred (stratResult g₁ D fuel strat) g₁)).card + ≤ (fuel + D + 4).choose 2 * D + (D + 1) := by + refine (winIndex_card_le hord₁ hp strat fuel).trans ?_ + -- (A) the δ = D all-pairs card bound, with (B)'s degree discharge on the real runTable + exact card_realWinSet_le_encoding_D strat (srsSt D) fuel D (fuel + D + 4) + (hdeg_out_of_run strat (srsSt D) fuel D (srsSt_table_natDegree_le D hD)) + (hdeg_handles_of_run strat (srsSt D) fuel D (srsSt_table_natDegree_le D hD)) + (by rw [srsSt_table_length]; omega) + -- lift the ℕ count bound through the ℝ≥0∞ division + exact ENNReal.div_le_div_right (by exact_mod_cast hcard) _ + +/-! ## 4. Real content: the bound is `< 1` in the standard regime. -/ + +omit [PrimeOrderWith G₂ p] in +/-- **`tSdh_ggm_sound_lt_one`.** Under the standard security regime +`C(fuel+D+4, 2)·D + (D+1) < p − 1`, the t-SDH advantage of `embed strat` is a genuine `< 1` — the +theorem has real content (it is not a restated `≤ 1`). Composing with `tSdh_ggm_sound` gives +`tSdhExperiment D (embed strat) < 1`. -/ +theorem tSdh_ggm_sound_lt_one + (hord₁ : orderOf g₁ = p) + (hD : 1 ≤ D) (strat : Strat p) (fuel : ℕ) + (hreg : (fuel + D + 4).choose 2 * D + (D + 1) < p - 1) : + tSdhExperiment (g₁ := g₁) (g₂ := g₂) D (embed g₁ D fuel strat) < 1 := by + refine lt_of_le_of_lt (tSdh_ggm_sound hord₁ hD strat fuel) ?_ + have hb0 : ((p - 1 : ℕ) : ℝ≥0∞) ≠ 0 := by + rw [Ne, Nat.cast_eq_zero]; omega + have hbtop : ((p - 1 : ℕ) : ℝ≥0∞) ≠ ⊤ := ENNReal.natCast_ne_top _ + rw [ENNReal.div_lt_iff (Or.inl hb0) (Or.inl hbtop), one_mul] + exact_mod_cast hreg + +/-! ## Axiom hygiene — the capstone rests on exactly `[propext, Classical.choice, Quot.sound]`. -/ + +#print axioms tSdh_ggm_sound +#print axioms tSdh_ggm_sound_lt_one + +end GgmEndToEnd diff --git a/ArkLib/Scratch/KzgVacuity/GgmProbThreading.lean b/ArkLib/Scratch/KzgVacuity/GgmProbThreading.lean new file mode 100644 index 0000000000..e01fb54e89 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmProbThreading.lean @@ -0,0 +1,183 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Commitments.Functional.KZG.HardnessAssumptions + +/-! +# Threading the $t$-SDH probability game to a finite cardinality count + +ArkLib's real `tSdhExperiment` is an `OptionT ProbComp` game whose adversary lives in +`StateT unifSpec.QueryCache ProbComp`. This file reduces it to a plain `Finset`-cardinality count +$\#\mathrm{winSet} / (p - 1)$ in $\mathbb{R}_{\ge 0}^{\infty}$, importing ArkLib's real +`Groups.tSdhGame`, `Groups.tSdhExperiment`, `Groups.tSdhCondition`, and +`Groups.sampleNonzeroZMod` without restating them. + +ArkLib's hardness game for the $t$-SDH assumption [BB04] underlying KZG [KZG10] is +`tSdhExperiment D A = Pr[tSdhCondition | tSdhGame D A]`, with +`tSdhGame D A = OptionT.mk (do τ ← sampleNonzeroZMod; …; (A srs).run' ∅; pure …)`, an +$\mathbb{R}_{\ge 0}^{\infty}$ probability over the `OptionT`/`StateT`/`ProbComp` monad stack. The +generic-group counting bounds (`GgmAdaptive`, `GgmRandomEncoding`, `GgmArkLibTransport`) are +$\mathbb{Q}$-cardinality statements about a set of winning trapdoors. This file threads the two +together for any adversary that is deterministic-given-$\tau$ from an empty cache — exactly the +shape the `embed` construction produces (`(embed strat srs).run' ∅ = pure result`): + +* `game_collapse`: peel the game monad. With + `(A (PowerSrs.generate D τ)).run' ∅ = pure (resultOf τ)`, the whole `OptionT`/`StateT` game + collapses to + `OptionT.mk (do τ ← sampleNonzeroZMod; pure ((resultOf τ).map (τ, ·, ·)))`. +* `experiment_eq_count`: count the sampler. That probability equals $\#\mathrm{winSet} / (p - 1)$, + where `winSet` is the set of nonzero-trapdoor indices `i : Fin (p-1)` on which the deterministic + adversary's output wins `tSdhCondition`. + +The reduction reuses VCVio's `probEvent_uniformSample` and `probEvent_map`, together with +`OptionT.probOutput_eq` / `probEvent_eq_tsum_subtype` for the `OptionT.mk` peel (the `none`-mass is +handled by a subtype reindexing, with no $\mathbb{R}_{\ge 0}^{\infty}$ subtraction). ArkLib's +`Binding.lean` is the precedent for the identical-shape +`Pr[· | OptionT.mk (do τ ← sampleNonzeroZMod; …)]` reduction. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Kate, A., Zaverucha, G. M., and Goldberg, I., *Constant-Size Commitments to Polynomials and + Their Applications*][KZG10] +-/ + +open OracleSpec OracleComp Groups +open scoped Classical NNReal ENNReal + +namespace Ggm.ProbThreading + +variable {p : ℕ} [Fact (Nat.Prime p)] + [∀ i, SampleableType (unifSpec.Range i)] + {G₁ : Type} [Group G₁] [PrimeOrderWith G₁ p] {g₁ : G₁} + {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] {g₂ : G₂} + +/-! ### The `OptionT.mk` peel + +A generic lemma: the success probability of an event `P` under an `OptionT ProbComp` +computation built from `M : ProbComp (Option α)` equals, in the plain `ProbComp`, the +probability that the sampled `Option` is `some a` with `P a`. Failure (`none`) never counts +toward the event. Proven by reindexing the event subtype through `a ↦ some a`, so there is no +`ℝ≥0∞` subtraction of the failure mass. -/ +omit [∀ i, SampleableType (unifSpec.Range i)] in +lemma probEvent_optionT_mk {α : Type} (M : ProbComp (Option α)) (P : α → Prop) : + Pr[P | (OptionT.mk M : OptionT ProbComp α)] + = Pr[fun o => ∃ a, o = some a ∧ P a | M] := by + rw [probEvent_eq_tsum_subtype (OptionT.mk M) P, + probEvent_eq_tsum_subtype M (fun o => ∃ a, o = some a ∧ P a)] + simp only [OptionT.probOutput_eq, OptionT.run_mk] + -- goal: ∑' x : {a // P a}, Pr[= some ↑x | M] = ∑' o : {o // ∃ a, o = some a ∧ P a}, Pr[= ↑o | M] + let e : {a : α // P a} ≃ {o : Option α // ∃ a, o = some a ∧ P a} := + { toFun := fun x => ⟨some x.1, x.1, rfl, x.2⟩ + invFun := fun o => ⟨o.2.choose, o.2.choose_spec.2⟩ + left_inv := by + intro x + apply Subtype.ext + have h := (Exists.choose_spec (⟨x.1, rfl, x.2⟩ : ∃ a, some x.1 = some a ∧ P a)).1 + exact (Option.some.inj h).symm + right_inv := by + intro o + apply Subtype.ext + exact (o.2.choose_spec).1.symm } + exact Equiv.tsum_eq e (fun o => Pr[= (↑o : Option α) | M]) + +/-! ### (c1) The game collapse + +For an adversary that is deterministic-given-τ from an empty cache, the whole +`OptionT`/`StateT`/`ProbComp` game monad collapses to the trapdoor sampler followed by a +pure deterministic tail. `resultOf τ` is the adversary's `Option (challenge, group elt)` +output on the SRS generated from `τ`. -/ + +/-- The collapsed game's underlying `ProbComp (Option _)`: sample the nonzero trapdoor, then +tag the deterministic output with it. -/ +noncomputable def collapsedGameRun (resultOf : ZMod p → Option (ZMod p × G₁)) : + ProbComp (Option (ZMod p × ZMod p × G₁)) := + sampleNonzeroZMod (p := p) >>= fun τ => + pure ((resultOf τ).map (fun ((c, h) : ZMod p × G₁) => (τ, c, h))) + +omit [PrimeOrderWith G₁ p] [PrimeOrderWith G₂ p] in +/-- **(c1) `game_collapse`.** Peel the game monad: with the adversary deterministic-given-τ +from the empty cache (`hdet`), `tSdhGame` equals `OptionT.mk` of the collapsed run. -/ +theorem game_collapse (D : ℕ) (A : tSdhAdversary D (G₁ := G₁) (G₂ := G₂) (p := p)) + (resultOf : ZMod p → Option (ZMod p × G₁)) + (hdet : ∀ τ, (A (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ)).run' ∅ + = pure (resultOf τ)) : + tSdhGame (g₁ := g₁) (g₂ := g₂) D A + = OptionT.mk (collapsedGameRun resultOf) := by + unfold tSdhGame collapsedGameRun + simp only [hdet, pure_bind] + +/-! ### (c2) Count the sampler + +The nonzero-trapdoor sampler is `(i ↦ (i+1 : ZMod p)) <$> $ᵗ(Fin (p-1))`, so any event over +it is `(filter).card / (p-1)` by `probEvent_map` + `probEvent_uniformSample`. -/ + +omit [∀ i, SampleableType (unifSpec.Range i)] in +/-- Every event over the nonzero-trapdoor sampler is a `Fin (p-1)` count over `p-1`. +`(i : Fin (p-1))` maps to trapdoor `(i+1 : ZMod p)`, ranging over the nonzero residues. -/ +lemma probEvent_sampleNonzeroZMod (q : ZMod p → Prop) : + Pr[q | (sampleNonzeroZMod (p := p))] + = ((Finset.univ.filter (fun i : Fin (p - 1) => q ((i : ℕ) + 1))).card : ℝ≥0∞) + / ((p - 1 : ℕ) : ℝ≥0∞) := by + haveI : NeZero (p - 1) := + ⟨Nat.pos_iff_ne_zero.mp (Nat.sub_pos_of_lt (Nat.Prime.one_lt Fact.out))⟩ + unfold sampleNonzeroZMod + rw [probEvent_map, probEvent_uniformSample, Fintype.card_fin] + rfl + +/-- The winning-trapdoor set: nonzero-trapdoor indices `i : Fin (p-1)` (trapdoor `i+1`) on +which the deterministic adversary outputs some `(c, h)` that satisfies ArkLib's +`tSdhCondition`. This is exactly the set whose cardinality the GGM/counting bounds bound. -/ +def winPred (resultOf : ZMod p → Option (ZMod p × G₁)) (g₁ : G₁) : + Fin (p - 1) → Prop := + fun i => ∃ ch : ZMod p × G₁, + resultOf ((i : ℕ) + 1 : ZMod p) = some ch ∧ + tSdhCondition (p := p) (g₁ := g₁) (((i : ℕ) + 1 : ZMod p), ch.1, ch.2) + +omit [PrimeOrderWith G₁ p] [PrimeOrderWith G₂ p] in +/-- **(c2) `experiment_eq_count`.** ArkLib's `tSdhExperiment` for a deterministic-given-τ +adversary equals the counting fraction `(winSet.card) / (p - 1)` in `ℝ≥0∞`. -/ +theorem experiment_eq_count (D : ℕ) (A : tSdhAdversary D (G₁ := G₁) (G₂ := G₂) (p := p)) + (resultOf : ZMod p → Option (ZMod p × G₁)) + (hdet : ∀ τ, (A (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ)).run' ∅ + = pure (resultOf τ)) : + tSdhExperiment (g₁ := g₁) (g₂ := g₂) D A + = ((Finset.univ.filter (winPred (p := p) resultOf g₁)).card : ℝ≥0∞) + / ((p - 1 : ℕ) : ℝ≥0∞) := by + unfold tSdhExperiment + rw [game_collapse (g₁ := g₁) (g₂ := g₂) D A resultOf hdet, + probEvent_optionT_mk (collapsedGameRun resultOf) + (tSdhCondition (p := p) (g₁ := g₁))] + unfold collapsedGameRun + rw [bind_pure_comp, probEvent_map] + -- Rewrite the composed event to the clean per-trapdoor winning predicate BEFORE counting. + -- (Doing it after `probEvent_sampleNonzeroZMod` would force a `DecidablePred` synthesis on the + -- composed `∃ … Option.map … tSdhCondition …` predicate, which loops.) + have hev : + Pr[((fun o => ∃ a, o = some a ∧ tSdhCondition (p := p) (g₁ := g₁) a) ∘ + fun τ => (resultOf τ).map (fun ((c, h) : ZMod p × G₁) => (τ, c, h))) + | sampleNonzeroZMod (p := p)] + = Pr[(fun τ => ∃ ch : ZMod p × G₁, resultOf τ = some ch ∧ + tSdhCondition (p := p) (g₁ := g₁) (τ, ch.1, ch.2)) + | sampleNonzeroZMod (p := p)] := by + apply probEvent_ext + intro τ _ + simp only [Function.comp_apply] + constructor + · rintro ⟨t, ht, hcond⟩ + rcases hopt : resultOf τ with _ | ch + · rw [hopt] at ht; simp at ht + · rw [hopt] at ht + simp only [Option.map_some, Option.some.injEq] at ht + exact ⟨ch, rfl, ht ▸ hcond⟩ + · rintro ⟨ch, hch, hcond⟩ + exact ⟨(τ, ch.1, ch.2), by rw [hch]; rfl, hcond⟩ + rw [hev, probEvent_sampleNonzeroZMod] + rfl + +end Ggm.ProbThreading + +#print axioms Ggm.ProbThreading.game_collapse +#print axioms Ggm.ProbThreading.experiment_eq_count diff --git a/ArkLib/Scratch/KzgVacuity/GgmRandomEncoding.lean b/ArkLib/Scratch/KzgVacuity/GgmRandomEncoding.lean new file mode 100644 index 0000000000..177bbad145 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmRandomEncoding.lean @@ -0,0 +1,573 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Scratch.KzgVacuity.GgmAdaptive +import Mathlib.Data.Sym.Card + +/-! +# All-pairs (random-encoding) collision bound for the generic-group t-SDH game + +This file strengthens the per-query adaptive generic-group-model bound of `GgmAdaptive` into an +all-pairs / global bad-event bound, in the style of the random-encoding analyses of the generic +group model [Sho97]. + +`GgmAdaptive` bounds the bad event *per explicit equality query*: `fuel` queries, each contributing +$\le \Delta$ collision trapdoors. Here that same explicit-query bad set is upper-bounded by the +global all-pairs collision set: some two formally-distinct table polynomials collide at the +trapdoor $\tau$. The resulting count is the conservative all-pairs number. + +The operational semantics remains `GgmAdaptive.Strat`, where comparisons must be requested with +`Move.query`. This file does not define a free-comparison random-encoding semantics or prove a +simulation theorem for one; it proves a sound, looser bound for the explicit-query embedded +strategies. The bound is parameterized by a per-handle degree $\Delta$ and instantiated at two +degrees: + +* $\Delta = D$ — the linear-oracle instantiation. The Boneh–Boyen t-SDH adversary [BB04] receives + `Vector G₁ (D+1) × Vector G₂ 2`, must output a `G₁` element, and is granted no pairing map + `e : G₁ × G₂ → Gₜ`. So every handle it can form is a `ZMod p`-linear combination of the seed + $\{1, X, \dots, X^D\}$, of degree $\le D$, and a difference of two such handles again has degree + $\le D$ (`natDegree_sub_le` — the max, not the sum). This is the $\sim (q+D)^2 D / p$ bound, with + the degree invariant discharged (not assumed) downstream. + +* $\Delta = 2D$ — an algebraic specialization (`rand_encoding_bound`, `rand_encoding_bound_srs`), + valid when the caller supplies a $\le 2D$ invariant for the same linear `runAux` semantics. It + does not model a pairing move; a genuine pairing-capable oracle would need a separate two-sorted + operational semantics and transport. + +Both instantiations are one-line specializations of the same general-$\Delta$ all-pairs lemma, so +the file has a single mathematical core; the two degrees differ only in the supplied handle-degree +hypothesis. + +## All-pairs Schwartz–Zippel + +For a finite set `ps` of polynomials of degree $\le \Delta$, the union over unordered pairs +$q_1 \ne q_2 \in$ `ps` of `roots (q₁ - q₂)` has card $\le \binom{\#ps}{2} \cdot \Delta$ +(`card_pairRootUnion_le`) [Sch80], [Zip79]. The unordered count is exact: +`roots (q₁ - q₂) = roots (q₂ - q₁)` as sets, so the ordered `offDiag` union is re-indexed through +`Sym2` (`Sym2.card_image_offDiag`), paying $\binom{n}{2}$, not $n(n-1)$. Degree bookkeeping: +`natDegree (q₁ - q₂) ≤ max (deg q₁) (deg q₂) ≤ Δ` (`natDegree_sub_le` — the max, not the sum: +differences of two degree-$\le \Delta$ handles stay $\le \Delta$, they do not compound to +$2\Delta$). + +## The handle table, structurally + +`runTable` is the final handle table of a run; every polynomial the adversary ever queried is in it +(or is the zero polynomial — the out-of-range handle default, i.e. the group identity), proven by +induction (`runAux_pairs_mem_runTable`), and its length is $\le$ (seed count) $+$ fuel, proven by +induction (`runTable_length_le`). With the SRS seeding (G₁: `1, X, …, X^D` = D+1 handles; G₂: `1, X` += 2 handles; seed count D+3) the handle-set card is $\le$ fuel $+ D + 4$ — the $+4$ is the $D+3$ +seeds plus the zero/identity handle. The table size is therefore a theorem here +(`card_handlePolys_le`), not a hypothesis. + +## The all-pairs bound + +Every deterministic `Strat` in the explicit-query model wins t-SDH on at most a +$(\binom{n}{2} \cdot \Delta + (D+1))/(p-1)$ fraction of trapdoors, for `n` any bound on the +handle-set size (`n = fuel + D + 4` at the SRS seeding), under the handle-degree invariant +$\le \Delta$. This is Shoup's $(q+d)^2 \Delta / p$ shape [Sho97] with the global bad event. + +The pieces reused from `GgmAdaptive` / `GgmCandidate` (not reproved): `realWinSet_subset` +(identical-until-bad at the set level), `card_winningPoints_le` (the static Boneh–Boyen root event +[BB04], behind which sit `winPoly_ne_zero`, `winPoly_natDegree_le`, `card_roots_winPoly_le`), +`badSet`/`badPolys`/`symPairs`, `adaptiveExperiment`. New here: the all-pairs union lemma, the +structural table lemmas, and the composition. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +* [Schwartz, J. T., *Fast Probabilistic Algorithms for Verification of Polynomial + Identities*][Sch80] +* [Zippel, R., *Probabilistic Algorithms for Sparse Polynomials*][Zip79] +-/ + +open Polynomial + +namespace GgmRandomEncoding + +open GgmCandidate GgmAdaptive + +variable {p : ℕ} [Fact (Nat.Prime p)] + +/-! ## § PairUnion — the all-pairs root-union bound + +The union, over all unordered pairs of distinct polynomials from a finite set, of the roots of the +pair's difference. Counting through `Sym2` pays `C(n,2)` — the ordered `offDiag` index would pay +`n(n−1)`, double, because `roots (q₁ − q₂)` and `roots (q₂ − q₁)` coincide as sets. -/ + +/-- The union of `roots (q₁ − q₂)` over all ordered pairs `q₁ ≠ q₂` of `ps` — which equals the +union over unordered pairs, since the two orders of a pair contribute the same root set. -/ +noncomputable def pairRootUnion (ps : Finset ((ZMod p)[X])) : Finset (ZMod p) := + ps.offDiag.biUnion fun q => (q.1 - q.2).roots.toFinset + +/-- Membership characterization of `pairRootUnion`: exactly the τ that are a root of the +difference of SOME pair of distinct polynomials from `ps`. -/ +theorem mem_pairRootUnion {ps : Finset ((ZMod p)[X])} {τ : ZMod p} : + τ ∈ pairRootUnion ps ↔ + ∃ q₁ q₂, q₁ ∈ ps ∧ q₂ ∈ ps ∧ q₁ ≠ q₂ ∧ τ ∈ (q₁ - q₂).roots.toFinset := by + unfold pairRootUnion + constructor + · intro h + obtain ⟨q, hq, hτ⟩ := Finset.mem_biUnion.mp h + obtain ⟨h1, h2, hne⟩ := Finset.mem_offDiag.mp hq + exact ⟨q.1, q.2, h1, h2, hne, hτ⟩ + · rintro ⟨q₁, q₂, h1, h2, hne, hτ⟩ + exact Finset.mem_biUnion.mpr ⟨(q₁, q₂), Finset.mem_offDiag.mpr ⟨h1, h2, hne⟩, hτ⟩ + +/-- The root set of the difference of an UNORDERED pair of polynomials — well-defined on `Sym2` +because `a − b` and `b − a` have the same roots (`a−b ≠ 0 ↔ b−a ≠ 0`, and the evaluations vanish +together). -/ +noncomputable def sym2DiffRoots : Sym2 ((ZMod p)[X]) → Finset (ZMod p) := + Sym2.lift ⟨fun a b => (a - b).roots.toFinset, by + intro a b + ext τ + simp only [Multiset.mem_toFinset, mem_roots', ne_eq, IsRoot.def, eval_sub, sub_eq_zero] + constructor + · rintro ⟨h1, h2⟩ + exact ⟨fun e => h1 e.symm, h2.symm⟩ + · rintro ⟨h1, h2⟩ + exact ⟨fun e => h1 e.symm, h2.symm⟩⟩ + +/-- `sym2DiffRoots` on a constructed pair is the difference's root set. -/ +lemma sym2DiffRoots_mk (a b : (ZMod p)[X]) : + sym2DiffRoots s(a, b) = (a - b).roots.toFinset := rfl + +/-- **ALL-PAIRS UNION SCHWARTZ–ZIPPEL.** If every polynomial in `ps` has degree ≤ Δ, the union of +the root sets of all pairwise differences has card ≤ `C(#ps, 2) · Δ`. The count is over UNORDERED +pairs (`Sym2.card_image_offDiag`); the degree of each difference is bounded by the MAX of the two +degrees (`natDegree_sub_le`), so a family of degree-≤Δ handles pays Δ per pair — never 2Δ. -/ +theorem card_pairRootUnion_le {ps : Finset ((ZMod p)[X])} {Δ : ℕ} + (hdeg : ∀ q ∈ ps, q.natDegree ≤ Δ) : + (pairRootUnion ps).card ≤ ps.card.choose 2 * Δ := by + classical + -- Re-index the ordered `offDiag` union through unordered pairs: same union, `C(n,2)` indices. + have hrw : pairRootUnion ps = (ps.offDiag.image Sym2.mk.uncurry).biUnion sym2DiffRoots := by + unfold pairRootUnion + rw [Finset.image_biUnion] + refine Finset.biUnion_congr rfl fun q _ => ?_ + obtain ⟨a, b⟩ := q + exact (sym2DiffRoots_mk a b).symm + rw [hrw] + refine Finset.card_biUnion_le.trans ?_ + calc ∑ s ∈ ps.offDiag.image Sym2.mk.uncurry, (sym2DiffRoots s).card + ≤ ∑ _s ∈ ps.offDiag.image Sym2.mk.uncurry, Δ := by + refine Finset.sum_le_sum fun s hs => ?_ + obtain ⟨⟨a, b⟩, hab, rfl⟩ := Finset.mem_image.mp hs + obtain ⟨ha, hb, -⟩ := Finset.mem_offDiag.mp hab + show ((a - b).roots.toFinset).card ≤ Δ + refine (Multiset.toFinset_card_le _).trans ((card_roots' _).trans ?_) + exact (natDegree_sub_le a b).trans (max_le (hdeg a ha) (hdeg b hb)) + _ = (ps.offDiag.image Sym2.mk.uncurry).card * Δ := by rw [Finset.sum_const, smul_eq_mul] + _ = ps.card.choose 2 * Δ := by rw [Sym2.card_image_offDiag] + +/-- The δ = 2D specialization: a set of degree-≤2D handle polynomials has all-pairs collision set of +card ≤ `C(n,2) · 2D`. Feeds the off-path conservative chain (PAPER §9.2). -/ +theorem card_pairRootUnion_le_two_mul {ps : Finset ((ZMod p)[X])} {D : ℕ} + (hdeg : ∀ q ∈ ps, q.natDegree ≤ 2 * D) : + (pairRootUnion ps).card ≤ ps.card.choose 2 * (2 * D) := + card_pairRootUnion_le hdeg + +/-! ## § Table — the run's handle table, structurally + +`runTable` mirrors `runAux`'s recursion on the table component and returns the FINAL handle table. +Three facts, each by induction on fuel: the initial table is a prefix of the final one; every +polynomial behind a queried handle is in the final table or is `0` (the out-of-range `getD` +default — the group identity's polynomial); and the final table grows by at most one entry per +fuel step. -/ + +/-- The final handle table of a run (the table component of `runAux`'s final state). -/ +noncomputable def runTable (ans : AnswerFn p) (strat : Strat p) : + ℕ → St p → List ((ZMod p)[X]) + | 0, st => st.table + | fuel + 1, st => + match strat st.hist with + | Sum.inr _ => st.table + | Sum.inl (Move.lin spec) => + runTable ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ + | Sum.inl (Move.query i j) => + runTable ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ + +/-- The table only ever grows: the current table is a prefix of the final one. -/ +theorem table_prefix_runTable (ans : AnswerFn p) (strat : Strat p) : + ∀ (fuel : ℕ) (st : St p), st.table <+: runTable ans strat fuel st := by + intro fuel + induction fuel with + | zero => intro st; simp only [runTable]; exact List.prefix_rfl + | succ fuel ih => + intro st + rcases hdec : strat st.hist with m | out + · cases m with + | lin spec => + have e : runTable ans strat (fuel + 1) st + = runTable ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runTable, hdec] + rw [e] + exact (List.prefix_append _ _).trans (ih ⟨st.table ++ [combine spec st.table], st.hist⟩) + | query i j => + have e : runTable ans strat (fuel + 1) st + = runTable ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ := by + simp only [runTable, hdec] + rw [e] + exact ih ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ + · have e : runTable ans strat (fuel + 1) st = st.table := by + simp only [runTable, hdec] + rw [e] + +/-- Each fuel step appends at most one handle: the final table has length ≤ initial + fuel. -/ +theorem runTable_length_le (ans : AnswerFn p) (strat : Strat p) : + ∀ (fuel : ℕ) (st : St p), + (runTable ans strat fuel st).length ≤ st.table.length + fuel := by + intro fuel + induction fuel with + | zero => intro st; simp only [runTable]; omega + | succ fuel ih => + intro st + rcases hdec : strat st.hist with m | out + · cases m with + | lin spec => + have e : runTable ans strat (fuel + 1) st + = runTable ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runTable, hdec] + rw [e] + have := ih ⟨st.table ++ [combine spec st.table], st.hist⟩ + simp only [List.length_append, List.length_cons, List.length_nil] at this + omega + | query i j => + have e : runTable ans strat (fuel + 1) st + = runTable ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ := by + simp only [runTable, hdec] + rw [e] + have := ih ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ + simp only at this + omega + · have e : runTable ans strat (fuel + 1) st = st.table := by + simp only [runTable, hdec] + rw [e] + omega + +/-- A defaulted list lookup is either a genuine element or the default. -/ +lemma getD_mem_or_eq_zero (l : List ((ZMod p)[X])) (i : ℕ) : + l.getD i 0 ∈ l ∨ l.getD i 0 = (0 : (ZMod p)[X]) := by + rw [List.getD_eq_getElem?_getD] + cases h : l[i]? with + | none => right; rfl + | some a => + left + have ha := List.mem_of_getElem? h + simpa using ha + +/-- **Every queried handle polynomial is in the final table (or is `0`).** By induction on fuel: +a query's two components are defaulted lookups into the CURRENT table, which is a prefix of the +final one; the tail pairs come from the recursive run over the same final table. -/ +theorem runAux_pairs_mem_runTable (ans : AnswerFn p) (strat : Strat p) : + ∀ (fuel : ℕ) (st : St p), ∀ ab ∈ (runAux ans strat fuel st).2, + (ab.1 ∈ runTable ans strat fuel st ∨ ab.1 = 0) ∧ + (ab.2 ∈ runTable ans strat fuel st ∨ ab.2 = 0) := by + intro fuel + induction fuel with + | zero => intro st ab hab; simp [runAux] at hab + | succ fuel ih => + intro st ab hab + rcases hdec : strat st.hist with m | out + · cases m with + | lin spec => + have e : runAux ans strat (fuel + 1) st + = runAux ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runAux, hdec] + have eT : runTable ans strat (fuel + 1) st + = runTable ans strat fuel ⟨st.table ++ [combine spec st.table], st.hist⟩ := by + simp only [runTable, hdec] + rw [e] at hab + rw [eT] + exact ih ⟨st.table ++ [combine spec st.table], st.hist⟩ ab hab + | query i j => + have e : runAux ans strat (fuel + 1) st + = ((runAux ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩).1, + (st.table.getD i 0, st.table.getD j 0) :: + (runAux ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩).2) := by + simp only [runAux, hdec] + have eT : runTable ans strat (fuel + 1) st + = runTable ans strat fuel + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ := by + simp only [runTable, hdec] + rw [e] at hab + rw [eT] + -- the recursive state: same table, extended history + set st' : St p := + ⟨st.table, st.hist ++ [ans (st.table.getD i 0) (st.table.getD j 0)]⟩ with hst' + rcases List.mem_cons.mp hab with hhd | htl + · -- the head pair: current-table lookups; current table is a prefix of the final one. + have hpre : st.table ⊆ runTable ans strat fuel st' := + (table_prefix_runTable ans strat fuel st').subset + constructor + · rcases getD_mem_or_eq_zero st.table i with h | h + · left; rw [hhd]; exact hpre h + · right; rw [hhd]; exact h + · rcases getD_mem_or_eq_zero st.table j with h | h + · left; rw [hhd]; exact hpre h + · right; rw [hhd]; exact h + · exact ih st' ab htl + · have e : runAux ans strat (fuel + 1) st = ((out.1, st.table.getD out.2 0), []) := by + simp only [runAux, hdec] + rw [e] at hab + simp at hab + +/-- The finite set of handle polynomials a run can ever compare: the final table plus the zero +polynomial (the identity handle backing out-of-range lookups). -/ +noncomputable def handlePolys (ans : AnswerFn p) (strat : Strat p) (fuel : ℕ) (st : St p) : + Finset ((ZMod p)[X]) := + insert 0 (runTable ans strat fuel st).toFinset + +/-- Membership in `handlePolys` from the disjunction `runAux_pairs_mem_runTable` produces. -/ +lemma mem_handlePolys_of {ans : AnswerFn p} {strat : Strat p} {fuel : ℕ} {st : St p} + {f : (ZMod p)[X]} (h : f ∈ runTable ans strat fuel st ∨ f = 0) : + f ∈ handlePolys ans strat fuel st := by + unfold handlePolys + rcases h with h | h + · exact Finset.mem_insert_of_mem (List.mem_toFinset.mpr h) + · rw [h]; exact Finset.mem_insert_self _ _ + +/-- **The table-size bound, a THEOREM.** The handle set has card ≤ (seed count) + fuel + 1: +one appended handle per fuel step, plus the zero/identity handle. -/ +theorem card_handlePolys_le (ans : AnswerFn p) (strat : Strat p) (fuel : ℕ) (st : St p) : + (handlePolys ans strat fuel st).card ≤ st.table.length + fuel + 1 := by + refine (Finset.card_insert_le _ _).trans ?_ + have h := (List.toFinset_card_le (runTable ans strat fuel st)).trans + (runTable_length_le ans strat fuel st) + omega + +/-! ## § Bound — the all-pairs adaptive bound at δ = 2D (OFF-PATH specialization) + +The Shoup bad set of the adaptive file collects roots of differences of QUERIED pairs; every +queried pair lives in `handlePolys`, so the bad set is inside the all-pairs collision set of the +handle table — a conservative global collision event. Composing with the reused +`realWinSet_subset` and the reused static bound `card_winningPoints_le` gives +`C(n,2)·2D + (D+1)` winning trapdoors. + +This δ = 2D chain is a specialization of the general degree bound for the same linear `runAux` +semantics. It does not add a pairing move. ArkLib's `tSdhAdversary` is granted no pairing map, so +the critical-path bound is the δ = D sibling below. -/ + +/-- The per-query bad set is contained in the all-pairs collision set of the handle table. -/ +theorem badSet_subset_pairRootUnion (strat : Strat p) (st₀ : St p) (fuel : ℕ) : + badSet strat st₀ fuel ⊆ pairRootUnion (handlePolys symAns strat fuel st₀) := by + intro τ hτ + unfold badSet rootUnion at hτ + obtain ⟨q, hq, hroot⟩ := Finset.mem_biUnion.mp hτ + unfold badPolys at hq + rw [List.mem_toFinset, List.mem_map] at hq + obtain ⟨ab, habf, rfl⟩ := hq + rw [List.mem_filter] at habf + obtain ⟨hab, hne'⟩ := habf + have hne : ab.1 ≠ ab.2 := of_decide_eq_true hne' + have hab' : ab ∈ (runAux symAns strat fuel st₀).2 := hab + have hmem := runAux_pairs_mem_runTable symAns strat fuel st₀ ab hab' + exact mem_pairRootUnion.mpr + ⟨ab.1, ab.2, mem_handlePolys_of hmem.1, mem_handlePolys_of hmem.2, hne, hroot⟩ + +/-- **The all-pairs counting bound.** Assuming every handle polynomial in the linear run has +degree ≤ 2D and the output-degree +invariant, the adaptive adversary wins on ≤ `C(#handles, 2)·2D + (D+1)` trapdoors. +Reuses `realWinSet_subset` and `card_winningPoints_le`; only the bad-set half is new. -/ +theorem card_realWinSet_le_allPairs (strat : Strat p) (st₀ : St p) (fuel : ℕ) (D : ℕ) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handlePolys symAns strat fuel st₀, q.natDegree ≤ 2 * D) : + (realWinSet strat st₀ fuel).card ≤ + (handlePolys symAns strat fuel st₀).card.choose 2 * (2 * D) + (D + 1) := by + classical + refine (Finset.card_le_card (realWinSet_subset strat st₀ fuel D hdeg_out)).trans ?_ + refine (Finset.card_union_le _ _).trans ?_ + have hbad : (badSet strat st₀ fuel).card ≤ + (handlePolys symAns strat fuel st₀).card.choose 2 * (2 * D) := + (Finset.card_le_card (badSet_subset_pairRootUnion strat st₀ fuel)).trans + (card_pairRootUnion_le_two_mul hdeg_handles) + exact Nat.add_le_add hbad (card_winningPoints_le _) + +/-- The counting bound at an abstract table-size bound `n`: any `n ≥ (seed count) + fuel + 1` +works, by monotonicity of `C(·, 2)`. -/ +theorem card_realWinSet_le_encoding (strat : Strat p) (st₀ : St p) (fuel : ℕ) (D n : ℕ) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handlePolys symAns strat fuel st₀, q.natDegree ≤ 2 * D) + (hn : st₀.table.length + fuel + 1 ≤ n) : + (realWinSet strat st₀ fuel).card ≤ n.choose 2 * (2 * D) + (D + 1) := by + refine (card_realWinSet_le_allPairs strat st₀ fuel D hdeg_out hdeg_handles).trans ?_ + have hcard : (handlePolys symAns strat fuel st₀).card ≤ n := + (card_handlePolys_le symAns strat fuel st₀).trans hn + exact Nat.add_le_add_right (Nat.mul_le_mul_right _ (Nat.choose_le_choose 2 hcard)) _ + +/-- **THE ALL-PAIRS SECURITY BOUND at δ = 2D (sorry-free; OFF-PATH specialization).** +Every deterministic strategy in the explicit-query model whose handle table stays within `n` +polynomials (a THEOREM at +`n = seeds + fuel + 1`, `card_handlePolys_le`) wins on at most a +`(C(n,2)·2D + (D+1))/(p−1)` fraction +of trapdoors — the global all-pairs collision event plus the static root event. This theorem assumes +the `≤ 2D` invariant; it does not provide pairing-capable semantics. -/ +theorem rand_encoding_bound (strat : Strat p) (st₀ : St p) (fuel : ℕ) (D n : ℕ) (hp : 2 ≤ p) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handlePolys symAns strat fuel st₀, q.natDegree ≤ 2 * D) + (hn : st₀.table.length + fuel + 1 ≤ n) : + adaptiveExperiment strat st₀ fuel ≤ + ((n.choose 2 * (2 * D) + (D + 1) : ℕ) : ℚ) / (p - 1) := by + unfold adaptiveExperiment + have hnum : ((realWinSet strat st₀ fuel).card : ℚ) + ≤ ((n.choose 2 * (2 * D) + (D + 1) : ℕ) : ℚ) := by + exact_mod_cast card_realWinSet_le_encoding strat st₀ fuel D n hdeg_out hdeg_handles hn + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by exact_mod_cast hp + linarith + gcongr + +omit [Fact (Nat.Prime p)] in +/-- **Strictness of the all-pairs bound.** Whenever `C(n,2)·2D + (D+1) < p − 1` the bound +is a genuine rational `< 1`: at cryptographic parameters (`p ≈ 2²⁵⁴`, `D ≈ 2²⁰`, `n ≈ 2⁶⁰`) +`C(n,2)·2D ≈ 2¹⁴⁰ ≪ p`, so the bound is `≈ 2⁻¹¹⁴`. -/ +theorem rand_encoding_bound_lt_one (D n : ℕ) + (hlt : n.choose 2 * (2 * D) + (D + 1) < p - 1) (hp : 2 ≤ p) : + ((n.choose 2 * (2 * D) + (D + 1) : ℕ) : ℚ) / (p - 1) < 1 := by + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by exact_mod_cast hp + linarith + rw [div_lt_one hden] + have h1 : ((n.choose 2 * (2 * D) + (D + 1) : ℕ) : ℚ) < ((p - 1 : ℕ) : ℚ) := by + exact_mod_cast hlt + have h2 : ((p - 1 : ℕ) : ℚ) = (p : ℚ) - 1 := by + have : (1 : ℕ) ≤ p := by omega + push_cast [Nat.cast_sub this]; ring + rw [h2] at h1; exact h1 + +/-! ## § SRS — the seeded instantiation, `n = fuel + D + 4` + +The adaptive file's SRS seeding: G₁ handles `1, X, …, X^D` (D+1 polynomials), G₂ handles `1, X` +(2 polynomials) — seed count D+3. With the zero/identity handle, the handle set has card +≤ fuel + D + 4, so the concrete numerator is `C(fuel + D + 4, 2)·2D + (D+1)`. -/ + +/-- The SRS-seeded initial state: table `1, X, …, X^D, 1, X`, empty history. -/ +noncomputable def srsSt (D : ℕ) : St p := + ⟨((List.range (D + 1)).map fun i => (X : (ZMod p)[X]) ^ i) ++ [1, X], []⟩ + +/-- The SRS seed count: `(D+1) + 2 = D + 3` handles. -/ +theorem srsSt_table_length (D : ℕ) : (srsSt (p := p) D).table.length = D + 3 := by + simp only [srsSt, List.length_append, List.length_map, List.length_range, + List.length_cons, List.length_nil] + +/-- **The strengthened bound at the SRS seeding**: table size `n = fuel + D + 4` (the D+3 SRS +seeds, one appended handle per fuel step, and the zero/identity handle), giving the concrete +`(C(fuel+D+4, 2)·2D + (D+1))/(p−1)` — the `(q+d)²·δ/p` Shoup shape for t-SDH. -/ +theorem rand_encoding_bound_srs (strat : Strat p) (fuel : ℕ) (D : ℕ) (hp : 2 ≤ p) + (hdeg_out : (symOutput strat (srsSt D) fuel).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handlePolys symAns strat fuel (srsSt D), q.natDegree ≤ 2 * D) : + adaptiveExperiment strat (srsSt D) fuel ≤ + (((fuel + D + 4).choose 2 * (2 * D) + (D + 1) : ℕ) : ℚ) / (p - 1) := by + refine rand_encoding_bound strat (srsSt D) fuel D (fuel + D + 4) hp hdeg_out hdeg_handles ?_ + rw [srsSt_table_length] + omega + +/-! ## § Bound at δ = D — the linear-oracle instantiation (THE ARKLIB CRITICAL PATH) + +This is the chain the end-to-end capstone consumes. The embedded adversary receives +`Vector G₁ (D+1) × Vector G₂ 2`, must output a `G₁` element, and is granted **no pairing map** +`e : G₁ × G₂ → Gₜ`. So — matching `GgmAdaptive`'s pairing-free `Move` — every handle it can form +is a `ZMod p`-linear combination of the seed `{1, X, …, X^D}`, degree ≤ D (never a product). The +honest collision degree is therefore **δ = D**, not 2D: a difference of two degree-≤D handles has +degree ≤ D (`natDegree_sub_le` — the max, not the sum). This section re-parametrizes the +*general-Δ* `card_pairRootUnion_le` at Δ = D. It is the exact `~(q+D)²·D/p` Shoup socket the +end-to-end theorem consumes (`rand_encoding_bound_srs_D` → `GgmDegreeDischarge` → +`GgmEndToEnd.tSdh_ggm_sound`), with the handle-degree invariant DISCHARGED there, not assumed. The +δ = 2D chain above is only the off-path +degree specialization. -/ + +/-- The δ = D instance of the all-pairs root-union bound: a set of degree-≤D handle polynomials has +all-pairs collision set of card ≤ `C(n,2) · D`. Direct from the general-Δ +`card_pairRootUnion_le`. -/ +theorem card_pairRootUnion_le_D {ps : Finset ((ZMod p)[X])} {D : ℕ} + (hdeg : ∀ q ∈ ps, q.natDegree ≤ D) : + (pairRootUnion ps).card ≤ ps.card.choose 2 * D := + card_pairRootUnion_le hdeg + +/-- **The all-pairs counting bound at δ = D.** Under the linear-oracle degree invariant (every +handle polynomial has degree ≤ D — no pairing, so no product term) and the output-degree invariant, +the adaptive adversary wins on ≤ `C(#handles, 2)·D + (D + 1)` trapdoors. δ = D sibling of +`card_realWinSet_le_allPairs`. -/ +theorem card_realWinSet_le_allPairs_D (strat : Strat p) (st₀ : St p) (fuel : ℕ) (D : ℕ) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handlePolys symAns strat fuel st₀, q.natDegree ≤ D) : + (realWinSet strat st₀ fuel).card ≤ + (handlePolys symAns strat fuel st₀).card.choose 2 * D + (D + 1) := by + classical + refine (Finset.card_le_card (realWinSet_subset strat st₀ fuel D hdeg_out)).trans ?_ + refine (Finset.card_union_le _ _).trans ?_ + have hbad : (badSet strat st₀ fuel).card ≤ + (handlePolys symAns strat fuel st₀).card.choose 2 * D := + (Finset.card_le_card (badSet_subset_pairRootUnion strat st₀ fuel)).trans + (card_pairRootUnion_le_D hdeg_handles) + exact Nat.add_le_add hbad (card_winningPoints_le _) + +/-- The δ = D counting bound at an abstract table-size bound `n`. δ = D sibling of +`card_realWinSet_le_encoding`. -/ +theorem card_realWinSet_le_encoding_D (strat : Strat p) (st₀ : St p) (fuel : ℕ) (D n : ℕ) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handlePolys symAns strat fuel st₀, q.natDegree ≤ D) + (hn : st₀.table.length + fuel + 1 ≤ n) : + (realWinSet strat st₀ fuel).card ≤ n.choose 2 * D + (D + 1) := by + refine (card_realWinSet_le_allPairs_D strat st₀ fuel D hdeg_out hdeg_handles).trans ?_ + have hcard : (handlePolys symAns strat fuel st₀).card ≤ n := + (card_handlePolys_le symAns strat fuel st₀).trans hn + exact Nat.add_le_add_right (Nat.mul_le_mul_right _ (Nat.choose_le_choose 2 hcard)) _ + +/-- **THE ALL-PAIRS GGM SECURITY BOUND AT δ = D (sorry-free; THE ARKLIB CRITICAL PATH).** +Every adaptive generic t-SDH adversary in the linear (pairing-free) oracle model whose handle table +stays within `n` polynomials wins on at most a `(C(n,2)·D + (D + 1))/(p − 1)` fraction of +trapdoors — the exact `~(q+D)²·D/p` Shoup socket for ArkLib's `tSdhExperiment`, whose adversary +cannot pair. This is the bound the end-to-end capstone consumes (via `GgmDegreeDischarge`); the +δ = 2D `rand_encoding_bound` is the off-path `≤ 2D` specialization. -/ +theorem rand_encoding_bound_D (strat : Strat p) (st₀ : St p) (fuel : ℕ) (D n : ℕ) (hp : 2 ≤ p) + (hdeg_out : (symOutput strat st₀ fuel).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handlePolys symAns strat fuel st₀, q.natDegree ≤ D) + (hn : st₀.table.length + fuel + 1 ≤ n) : + adaptiveExperiment strat st₀ fuel ≤ + ((n.choose 2 * D + (D + 1) : ℕ) : ℚ) / (p - 1) := by + unfold adaptiveExperiment + have hnum : ((realWinSet strat st₀ fuel).card : ℚ) + ≤ ((n.choose 2 * D + (D + 1) : ℕ) : ℚ) := by + exact_mod_cast card_realWinSet_le_encoding_D strat st₀ fuel D n hdeg_out hdeg_handles hn + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by exact_mod_cast hp + linarith + gcongr + +/-- **The δ = D bound at the SRS seeding**: table size `n = fuel + D + 4`, giving the concrete +`(C(fuel+D+4, 2)·D + (D + 1))/(p − 1)`. δ = D sibling of `rand_encoding_bound_srs`. -/ +theorem rand_encoding_bound_srs_D (strat : Strat p) (fuel : ℕ) (D : ℕ) (hp : 2 ≤ p) + (hdeg_out : (symOutput strat (srsSt D) fuel).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handlePolys symAns strat fuel (srsSt D), q.natDegree ≤ D) : + adaptiveExperiment strat (srsSt D) fuel ≤ + (((fuel + D + 4).choose 2 * D + (D + 1) : ℕ) : ℚ) / (p - 1) := by + refine rand_encoding_bound_D strat (srsSt D) fuel D (fuel + D + 4) hp hdeg_out hdeg_handles ?_ + rw [srsSt_table_length] + omega + +end GgmRandomEncoding + +-- Axiom receipts: every headline theorem is sorry-free on the standard three axioms. +#print axioms GgmRandomEncoding.card_pairRootUnion_le +#print axioms GgmRandomEncoding.card_pairRootUnion_le_two_mul +#print axioms GgmRandomEncoding.runAux_pairs_mem_runTable +#print axioms GgmRandomEncoding.card_handlePolys_le +#print axioms GgmRandomEncoding.badSet_subset_pairRootUnion +#print axioms GgmRandomEncoding.card_realWinSet_le_allPairs +#print axioms GgmRandomEncoding.card_realWinSet_le_encoding +#print axioms GgmRandomEncoding.rand_encoding_bound +#print axioms GgmRandomEncoding.rand_encoding_bound_lt_one +#print axioms GgmRandomEncoding.rand_encoding_bound_srs +#print axioms GgmRandomEncoding.card_pairRootUnion_le_D +#print axioms GgmRandomEncoding.card_realWinSet_le_allPairs_D +#print axioms GgmRandomEncoding.card_realWinSet_le_encoding_D +#print axioms GgmRandomEncoding.rand_encoding_bound_D +#print axioms GgmRandomEncoding.rand_encoding_bound_srs_D diff --git a/ArkLib/Scratch/KzgVacuity/GgmShoup.lean b/ArkLib/Scratch/KzgVacuity/GgmShoup.lean new file mode 100644 index 0000000000..8ce12a7275 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmShoup.lean @@ -0,0 +1,496 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Scratch.KzgVacuity.GgmDegreeDischarge + +/-! +# The Shoup random-encoding GGM $t$-SDH bound with a free-comparison adversary + +This file mechanizes the random-encoding (Shoup) generic-group model for the $t$-SDH problem +[Sho97], [BB04] — the second standard GGM track, alongside the Maurer explicit-equality track +[Mau05]. In Maurer's model the adversary must spend a query step to test one handle pair, and only +queried pairs enter the bad event. Here the model is genuinely different: comparison is free. + +## The Shoup random-encoding model + +Group elements are lazily-sampled encodings under an injection +$\sigma : \mathbb{Z}_p \hookrightarrow E$. The adversary holds a `List E`, applies the group-op +oracle (`lin` — degree $\le D$, pairing-free, matching ArkLib's $G_1$-only `tSdhAdversary`), and — +the crux distinction from Maurer — may compare any two encodings it holds for free via +`DecidableEq E`. So at every step it observes the full $|tbl| \times |tbl|$ equality pattern of all +its held handles at zero fuel and handle cost, and branches on the entire pattern-history. + +Because $\sigma$ is injective, the equality pattern it observes, +$\sigma(f_i(\tau)) = \sigma(f_j(\tau)) \iff f_i(\tau) = f_j(\tau)$, is exactly the eval-at-$\tau$ +pattern (`realAns τ`); the symbolic ($\tau$-independent) pattern is formal equality (`symAns`). So +the encoding $\sigma$ never enters the mechanization — injectivity folds it away, exactly as +`GgmArkLibTransport.gpow_val_inj_iff` folds the concrete encoding $a \mapsto g^{a.\mathrm{val}}$ +away in the Maurer embed. $E$ and $\sigma$ live only in this docstring, where they name the model. + +Free comparison makes the all-pairs collision event tight: the adversary really can compare any +pair, so the leak is on exactly `pairRootUnion (handleSet)` — no $\subseteq$ slack, no `badSet` +indirection. + +## Structure + +* The model: `eqPattern` (the free full-table equality matrix, `Σ n, Fin n → Fin n → Bool`), + `ShoupMove` (`lin`-only — no query, equality is ambient), `ShoupStrat` (decides on the + pattern-history), `ShSt`, `runShoup`, `realWinSetShoup`, `shoupExperiment`. +* The hybrid, the crux, proven not assumed: `runShoup_congr_off_bad` — a matrix-valued + identical-until-bad. If $\tau \notin$ `pairRootUnion (handleSet)` then the real run equals the + symbolic run, with step-wise full-pattern agreement discharged from the single global + non-collision fact (every reachable table is a subset of the final handle set). This is the + free-comparison analogue of `GgmAdaptive.runAux_congr_of_agree`, but the branching input is a + whole equality matrix, and the bad event is the all-pairs union. +* The bound: `realWinSetShoup ⊆ pairRootUnion (handleSet) ∪ winningPoints sym`, composed with the + reused `GgmRandomEncoding.card_pairRootUnion_le_D` (all-pairs Schwartz–Zippel [Sch80], [Zip79]) + and `GgmCandidate.card_winningPoints_le` (static Boneh–Boyen root event [BB04]), giving numerator + $\binom{fuel+D+4}{2} \cdot D + (D+1)$. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +* [Maurer, U., *Abstract Models of Computation in Cryptography*][Mau05] +* [Schwartz, J. T., *Fast Probabilistic Algorithms for Verification of Polynomial + Identities*][Sch80] +* [Zippel, R., *Probabilistic Algorithms for Sparse Polynomials*][Zip79] +-/ + +open Polynomial + +namespace GgmShoup + +open GgmCandidate GgmAdaptive GgmRandomEncoding GgmDegreeDischarge GgmDegreeInvariant + +variable {p : ℕ} [Fact (Nat.Prime p)] + +/-! ## § Model — the free-comparison random-encoding oracle + +The crux distinction from Maurer. Comparison is AMBIENT and FREE: at every step the adversary +observes the full equality pattern (the pairwise-equality matrix under the oracle) of ALL handles it +holds, at no fuel and no handle cost, and branches on the entire pattern-history. There is no +`query` move — equality is not a spent step. `lin` is the only oracle-consuming move (degree ≤ D, +pairing-free: ArkLib's `tSdhAdversary` is granted no pairing map). -/ + +/-- The free-comparison observation: the `|tbl|×|tbl|` equality matrix under the oracle `ans`. +Packaged with its dimension so successive (growing) observations have a uniform type. -/ +noncomputable def eqPattern (ans : AnswerFn p) (tbl : List ((ZMod p)[X])) : + Σ n : ℕ, Fin n → Fin n → Bool := + ⟨tbl.length, fun i j => ans (tbl.get i) (tbl.get j)⟩ + +/-- A Shoup (random-encoding) group-op move: `lin` appends `Σ cᵢ · handleᵢ`. There is NO `query` +constructor — equality is ambient (see `eqPattern`), never a spent step. `lin` is pairing-free +(degree ≤ D), matching ArkLib's G₁-only `tSdhAdversary`. -/ +inductive ShoupMove (p : ℕ) where + | lin : List (ZMod p × ℕ) → ShoupMove p + +/-- A Shoup (random-encoding) strategy: decides on the HISTORY OF FULL EQUALITY PATTERNS observed so +far — never on a chosen single query. Equality is ambient, so the input is a `List` of full-table +patterns, one per step, not a `List Bool` of chosen-pair answers (the Maurer input). This typing is +what makes the model random-encoding-and-free-comparison rather than Maurer. -/ +abbrev ShoupStrat (p : ℕ) := List (Σ n : ℕ, Fin n → Fin n → Bool) → ShoupMove p ⊕ (ZMod p × ℕ) + +/-- Shoup oracle state: the handle table and the pattern-history. -/ +structure ShSt (p : ℕ) where + table : List ((ZMod p)[X]) + phist : List (Σ n : ℕ, Fin n → Fin n → Bool) + +/-- **The Shoup run.** Each step: observe the current table's FULL equality pattern under `ans` +(free comparison), append it to the pattern-history, let `strat` decide. `lin` appends a `combine`; +output reads out `(offset, table.getD k 0)`. Mirrors `GgmAdaptive.runAux` but threads PATTERNS, not +booleans, and has no `query` branch. -/ +noncomputable def runShoup (ans : AnswerFn p) (strat : ShoupStrat p) : + ℕ → ShSt p → (ZMod p × (ZMod p)[X]) + | 0, _ => (0, 0) + | fuel + 1, st => + match strat (st.phist ++ [eqPattern ans st.table]) with + | Sum.inr (c, k) => (c, st.table.getD k 0) + | Sum.inl (ShoupMove.lin spec) => + runShoup ans strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern ans st.table]⟩ + +/-- The final handle table of a Shoup run (mirrors `GgmRandomEncoding.runTable`, `lin`-only). -/ +noncomputable def runTableShoup (ans : AnswerFn p) (strat : ShoupStrat p) : + ℕ → ShSt p → List ((ZMod p)[X]) + | 0, st => st.table + | fuel + 1, st => + match strat (st.phist ++ [eqPattern ans st.table]) with + | Sum.inr _ => st.table + | Sum.inl (ShoupMove.lin spec) => + runTableShoup ans strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern ans st.table]⟩ + +/-- The finite set of handle polynomials the SYMBOLIC run can ever hold: the final symbolic table +plus the zero/identity handle (the out-of-range `getD` default). This is the set whose all-pairs +collision event is TIGHT for the free-comparison adversary. -/ +noncomputable def handleSetShoup (strat : ShoupStrat p) (st₀ : ShSt p) (fuel : ℕ) : + Finset ((ZMod p)[X]) := + insert 0 (runTableShoup symAns strat fuel st₀).toFinset + +/-- The trapdoors on which the free-comparison adversary wins t-SDH against the REAL encoding oracle +at `τ`. Same τ+c≠0-guarded win predicate reused verbatim from the static/Maurer files. -/ +noncomputable def realWinSetShoup (strat : ShoupStrat p) (st₀ : ShSt p) (fuel : ℕ) : + Finset (ZMod p) := + nonzeroPoints.filter (fun τ => + τ + (runShoup (realAns τ) strat fuel st₀).1 ≠ 0 ∧ + (runShoup (realAns τ) strat fuel st₀).2.eval τ + = 1 / (τ + (runShoup (realAns τ) strat fuel st₀).1)) + +/-- The Shoup success fraction: winning trapdoors over the `p−1` nonzero trapdoors. -/ +noncomputable def shoupExperiment (strat : ShoupStrat p) (st₀ : ShSt p) (fuel : ℕ) : ℚ := + (realWinSetShoup strat st₀ fuel).card / (p - 1) + +/-! ## § Table — structural lemmas for the Shoup handle table + +`runTableShoup` only grows the table (one `combine` per `lin` step), so its final table has the +current table as a prefix and length ≤ initial + fuel. Both by induction on fuel. -/ + +/-- The table only ever grows: the current table is a prefix of the final one. -/ +theorem table_prefix_runTableShoup (ans : AnswerFn p) (strat : ShoupStrat p) : + ∀ (fuel : ℕ) (st : ShSt p), st.table <+: runTableShoup ans strat fuel st := by + intro fuel + induction fuel with + | zero => intro st; simp only [runTableShoup]; exact List.prefix_rfl + | succ fuel ih => + intro st + rcases hdec : strat (st.phist ++ [eqPattern ans st.table]) with m | out + · cases m with + | lin spec => + have e : runTableShoup ans strat (fuel + 1) st + = runTableShoup ans strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern ans st.table]⟩ := by + simp only [runTableShoup, hdec] + rw [e] + exact (List.prefix_append _ _).trans + (ih ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern ans st.table]⟩) + · have e : runTableShoup ans strat (fuel + 1) st = st.table := by + simp only [runTableShoup, hdec] + rw [e] + +/-- Each fuel step appends at most one handle: the final table has length ≤ initial + fuel. -/ +theorem runTableShoup_length_le (ans : AnswerFn p) (strat : ShoupStrat p) : + ∀ (fuel : ℕ) (st : ShSt p), + (runTableShoup ans strat fuel st).length ≤ st.table.length + fuel := by + intro fuel + induction fuel with + | zero => intro st; simp only [runTableShoup]; omega + | succ fuel ih => + intro st + rcases hdec : strat (st.phist ++ [eqPattern ans st.table]) with m | out + · cases m with + | lin spec => + have e : runTableShoup ans strat (fuel + 1) st + = runTableShoup ans strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern ans st.table]⟩ := by + simp only [runTableShoup, hdec] + rw [e] + have := ih ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern ans st.table]⟩ + simp only [List.length_append, List.length_cons, List.length_nil] at this + omega + · have e : runTableShoup ans strat (fuel + 1) st = st.table := by + simp only [runTableShoup, hdec] + rw [e]; omega + +/-- **The handle-set size bound, a THEOREM.** The symbolic handle set has card ≤ (seed count) + +fuel + 1: one appended handle per fuel step, plus the zero/identity handle. -/ +theorem card_handleSetShoup_le (strat : ShoupStrat p) (st₀ : ShSt p) (fuel : ℕ) : + (handleSetShoup strat st₀ fuel).card ≤ st₀.table.length + fuel + 1 := by + refine (Finset.card_insert_le _ _).trans ?_ + have h := (List.toFinset_card_le (runTableShoup symAns strat fuel st₀)).trans + (runTableShoup_length_le symAns strat fuel st₀) + omega + +/-! ## § Degree — the Shoup handle table and output stay degree-bounded + +The oracle is purely LINEAR (`lin` only): a `combine` over a degree-≤D table stays ≤ D +(`GgmDegreeDischarge.natDegree_combine_le`, REUSED). By induction on fuel, so is the whole final +table, and the committed output (a defaulted table read). -/ + +/-- **The degree invariant of the Shoup handle table.** If every seed polynomial has natDegree ≤ D, +so does every polynomial in the run's final table — for ANY answer function. -/ +theorem runTableShoup_natDegree_le (ans : AnswerFn p) (strat : ShoupStrat p) {D : ℕ} : + ∀ (fuel : ℕ) (st : ShSt p), (∀ q ∈ st.table, q.natDegree ≤ D) → + ∀ q ∈ runTableShoup ans strat fuel st, q.natDegree ≤ D := by + intro fuel + induction fuel with + | zero => intro st hst q hq; simp only [runTableShoup] at hq; exact hst q hq + | succ fuel ih => + intro st hst q hq + rcases hdec : strat (st.phist ++ [eqPattern ans st.table]) with m | out + · cases m with + | lin spec => + have e : runTableShoup ans strat (fuel + 1) st + = runTableShoup ans strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern ans st.table]⟩ := by + simp only [runTableShoup, hdec] + rw [e] at hq + refine ih _ ?_ q hq + intro r hr + rcases List.mem_append.mp hr with h | h + · exact hst r h + · rw [List.mem_singleton.mp h]; exact natDegree_combine_le hst spec + · have e : runTableShoup ans strat (fuel + 1) st = st.table := by + simp only [runTableShoup, hdec] + rw [e] at hq; exact hst q hq + +/-- **The handle-set degree invariant** — the zero/identity handle has degree 0. -/ +theorem handleSetShoup_natDegree_le (strat : ShoupStrat p) (st₀ : ShSt p) (fuel : ℕ) {D : ℕ} + (hseed : ∀ q ∈ st₀.table, q.natDegree ≤ D) : + ∀ q ∈ handleSetShoup strat st₀ fuel, q.natDegree ≤ D := by + intro q hq + unfold handleSetShoup at hq + rcases Finset.mem_insert.mp hq with h | h + · rw [h]; simp + · exact runTableShoup_natDegree_le symAns strat fuel st₀ hseed q (List.mem_toFinset.mp h) + +/-- **The committed output has degree ≤ D** — a defaulted table read at commit time (or `0` on +fuel exhaustion); every intermediate table is bounded. Induction on fuel, for ANY answer +function. -/ +theorem runShoup_output_natDegree_le (ans : AnswerFn p) (strat : ShoupStrat p) {D : ℕ} : + ∀ (fuel : ℕ) (st : ShSt p), (∀ q ∈ st.table, q.natDegree ≤ D) → + (runShoup ans strat fuel st).2.natDegree ≤ D := by + intro fuel + induction fuel with + | zero => intro st _; simp [runShoup] + | succ fuel ih => + intro st hst + rcases hdec : strat (st.phist ++ [eqPattern ans st.table]) with m | out + · cases m with + | lin spec => + have e : runShoup ans strat (fuel + 1) st + = runShoup ans strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern ans st.table]⟩ := by + simp only [runShoup, hdec] + rw [e] + refine ih _ ?_ + intro r hr + rcases List.mem_append.mp hr with h | h + · exact hst r h + · rw [List.mem_singleton.mp h]; exact natDegree_combine_le hst spec + · have e : runShoup ans strat (fuel + 1) st = (out.1, st.table.getD out.2 0) := by + simp only [runShoup, hdec] + rw [e]; exact natDegree_getD_le hst out.2 + +/-! ## § Hybrid — the matrix-valued identical-until-collision (the crux, S4) + +Shoup's identical-until-bad for the free-comparison model. Where Maurer's `runAux_congr_of_agree` +threads a per-query boolean agreement, here the branching input is a WHOLE equality matrix, and the +single global hypothesis `τ ∉ pairRootUnion(handleSet)` discharges every step's full-pattern +agreement — which is exactly WHY the bad event is all-pairs. -/ + +/-- Two answer functions agreeing entrywise on a table produce the SAME `eqPattern` +(same dimension by construction; the matrices coincide by `funext`). -/ +lemma eqPattern_congr {ans1 ans2 : AnswerFn p} {tbl : List ((ZMod p)[X])} + (h : ∀ i j : Fin tbl.length, ans1 (tbl.get i) (tbl.get j) = ans2 (tbl.get i) (tbl.get j)) : + eqPattern ans1 tbl = eqPattern ans2 tbl := by + unfold eqPattern + refine Sigma.ext rfl ?_ + refine heq_of_eq (funext fun i => funext fun j => ?_) + exact h i j + +/-- `pairRootUnion` is monotone: a larger handle set has a larger all-pairs collision set. -/ +lemma pairRootUnion_mono {s t : Finset ((ZMod p)[X])} (h : s ⊆ t) : + pairRootUnion s ⊆ pairRootUnion t := by + intro τ hτ + rw [mem_pairRootUnion] at hτ ⊢ + obtain ⟨q₁, q₂, h1, h2, hne, hr⟩ := hτ + exact ⟨q₁, q₂, h h1, h h2, hne, hr⟩ + +/-- **Pattern agreement off the all-pairs bad set.** For a table all of whose entries lie in a +handle set with `τ ∉ pairRootUnion`, the real (eval-at-τ) equality pattern equals the symbolic +(formal-equality) one: distinct entries do not collide at τ (τ not a root of their difference), +and equal ones agree trivially. -/ +lemma eqPattern_realAns_eq_symAns {τ : ZMod p} {tbl : List ((ZMod p)[X])} + (hτ : τ ∉ pairRootUnion tbl.toFinset) : + eqPattern (realAns τ) tbl = eqPattern symAns tbl := by + refine eqPattern_congr fun i j => ?_ + by_cases hfg : tbl.get i = tbl.get j + · simp only [realAns, symAns, hfg] + · have hne : tbl.get i - tbl.get j ≠ 0 := sub_ne_zero.mpr hfg + have hmem_i : tbl.get i ∈ tbl.toFinset := List.mem_toFinset.mpr (List.get_mem _ _) + have hmem_j : tbl.get j ∈ tbl.toFinset := List.mem_toFinset.mpr (List.get_mem _ _) + have hnotroot : τ ∉ (tbl.get i - tbl.get j).roots.toFinset := fun hc => + hτ (mem_pairRootUnion.mpr ⟨_, _, hmem_i, hmem_j, hfg, hc⟩) + have hevalne : (tbl.get i).eval τ ≠ (tbl.get j).eval τ := by + intro hE + apply hnotroot + rw [Multiset.mem_toFinset, mem_roots hne] + simp only [IsRoot.def, eval_sub, hE, sub_self] + simp only [realAns, symAns, decide_eq_decide] + exact ⟨fun h => absurd h hevalne, fun h => absurd h hfg⟩ + +/-- **IDENTICAL-UNTIL-BAD, the crux (matrix-valued), PROVEN not assumed.** If +`τ ∉ pairRootUnion(handleSet)` (the all-pairs collision set of the final SYMBOLIC handle table), +the real run and the symbolic run COINCIDE. Induction on fuel: at each step the two runs are in +lockstep so observe the same table; by `eqPattern_realAns_eq_symAns` the two FULL patterns are equal +(the current table is a prefix of the final handle set, and `pairRootUnion` is monotone), so `strat` +makes the same decision and the recursion continues in lockstep. The single global non-collision +hypothesis discharges every step's full-pattern agreement — the free-comparison analogue of +`GgmAdaptive.runAux_congr_of_agree`. -/ +theorem runShoup_congr_off_bad (strat : ShoupStrat p) {τ : ZMod p} : + ∀ (fuel : ℕ) (st : ShSt p), + τ ∉ pairRootUnion (insert 0 (runTableShoup symAns strat fuel st).toFinset) → + runShoup (realAns τ) strat fuel st = runShoup symAns strat fuel st := by + intro fuel + induction fuel with + | zero => intro st _; rfl + | succ fuel ih => + intro st hτ + -- The current table's entries all lie in the final symbolic handle set. + have hsub : st.table.toFinset + ⊆ insert 0 (runTableShoup symAns strat (fuel + 1) st).toFinset := by + have hpre : st.table <+: runTableShoup symAns strat (fuel + 1) st := + table_prefix_runTableShoup symAns strat (fuel + 1) st + intro x hx + exact Finset.mem_insert_of_mem (List.mem_toFinset.mpr (hpre.subset (List.mem_toFinset.mp hx))) + -- Hence the real and symbolic observations of the current table agree. + have hobs : eqPattern (realAns τ) st.table = eqPattern symAns st.table := + eqPattern_realAns_eq_symAns (fun hc => hτ (pairRootUnion_mono hsub hc)) + rcases hdec : strat (st.phist ++ [eqPattern symAns st.table]) with m | out + · cases m with + | lin spec => + have eR : runShoup (realAns τ) strat (fuel + 1) st + = runShoup (realAns τ) strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern symAns st.table]⟩ := by + simp only [runShoup, hobs, hdec] + have eS : runShoup symAns strat (fuel + 1) st + = runShoup symAns strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern symAns st.table]⟩ := by + simp only [runShoup, hdec] + have eT : runTableShoup symAns strat (fuel + 1) st + = runTableShoup symAns strat fuel + ⟨st.table ++ [combine spec st.table], st.phist ++ [eqPattern symAns st.table]⟩ := by + simp only [runTableShoup, hdec] + rw [eR, eS] + exact ih _ (by rw [← eT]; exact hτ) + · have eR : runShoup (realAns τ) strat (fuel + 1) st = (out.1, st.table.getD out.2 0) := by + simp only [runShoup, hobs, hdec] + have eS : runShoup symAns strat (fuel + 1) st = (out.1, st.table.getD out.2 0) := by + simp only [runShoup, hdec] + rw [eR, eS] + +/-- **Identical-until-bad, at the set level** (S5). Every real winning trapdoor either triggers the +all-pairs collision event or is a static win of the τ-independent symbolic output — Shoup's +`W₀ ⊆ pairRootUnion(handleSet) ∪ F`, with the bad set already EQUAL to `pairRootUnion(handleSet)` +(no `badSet` indirection: free comparison makes the all-pairs union primitive). -/ +theorem realWinSetShoup_subset (strat : ShoupStrat p) (st₀ : ShSt p) (fuel D : ℕ) + (hdeg_out : (runShoup symAns strat fuel st₀).2.natDegree ≤ D) : + realWinSetShoup strat st₀ fuel ⊆ + pairRootUnion (handleSetShoup strat st₀ fuel) ∪ + GgmCandidate.winningPoints + (⟨(runShoup symAns strat fuel st₀).1, (runShoup symAns strat fuel st₀).2, hdeg_out⟩ : + GenericAdversary D p) := by + classical + intro τ hτ + rw [realWinSetShoup, Finset.mem_filter] at hτ + obtain ⟨hnz, hcond1, hcond2⟩ := hτ + by_cases hbad : τ ∈ pairRootUnion (handleSetShoup strat st₀ fuel) + · exact Finset.mem_union_left _ hbad + · refine Finset.mem_union_right _ ?_ + have heq := runShoup_congr_off_bad strat fuel st₀ hbad + rw [heq] at hcond1 hcond2 + rw [GgmCandidate.winningPoints, Finset.mem_filter] + exact ⟨hnz, hcond1, hcond2⟩ + +/-! ## § Bound — compose the REUSED all-pairs SZ core with the static root event + +`Pr[win] ≤ Pr[bad] + Pr[sym win]`. `Pr[bad]` consumes `card_pairRootUnion_le_D` DIRECTLY (tight — +no `badSet` step); `Pr[sym win]` consumes `card_winningPoints_le`. Both REUSED verbatim. The degree +hypotheses are DISCHARGED via `handleSetShoup_natDegree_le` / `runShoup_output_natDegree_le`. -/ + +/-- **The all-pairs counting bound.** With the handle-set and output degree invariants (both +DISCHARGED below at the SRS seeding), the free-comparison adversary wins on ≤ `C(#handles, 2)·D + +(D+1)` trapdoors. Reuses `card_pairRootUnion_le_D` and `card_winningPoints_le`. -/ +theorem card_realWinSetShoup_le_allPairs (strat : ShoupStrat p) (st₀ : ShSt p) (fuel D : ℕ) + (hdeg_out : (runShoup symAns strat fuel st₀).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handleSetShoup strat st₀ fuel, q.natDegree ≤ D) : + (realWinSetShoup strat st₀ fuel).card ≤ + (handleSetShoup strat st₀ fuel).card.choose 2 * D + (D + 1) := by + classical + refine (Finset.card_le_card (realWinSetShoup_subset strat st₀ fuel D hdeg_out)).trans ?_ + refine (Finset.card_union_le _ _).trans ?_ + have hbad : (pairRootUnion (handleSetShoup strat st₀ fuel)).card ≤ + (handleSetShoup strat st₀ fuel).card.choose 2 * D := + card_pairRootUnion_le_D hdeg_handles + exact Nat.add_le_add hbad (card_winningPoints_le _) + +/-- The counting bound at an abstract table-size bound `n` (`C(·, 2)` monotone). -/ +theorem card_realWinSetShoup_le_encoding (strat : ShoupStrat p) (st₀ : ShSt p) (fuel D n : ℕ) + (hdeg_out : (runShoup symAns strat fuel st₀).2.natDegree ≤ D) + (hdeg_handles : ∀ q ∈ handleSetShoup strat st₀ fuel, q.natDegree ≤ D) + (hn : st₀.table.length + fuel + 1 ≤ n) : + (realWinSetShoup strat st₀ fuel).card ≤ n.choose 2 * D + (D + 1) := by + refine (card_realWinSetShoup_le_allPairs strat st₀ fuel D hdeg_out hdeg_handles).trans ?_ + have hcard : (handleSetShoup strat st₀ fuel).card ≤ n := + (card_handleSetShoup_le strat st₀ fuel).trans hn + exact Nat.add_le_add_right (Nat.mul_le_mul_right _ (Nat.choose_le_choose 2 hcard)) _ + +/-! ## § SRS — the seeded instantiation, `n = fuel + D + 4` + +REUSE `GgmRandomEncoding.srsSt`'s table (`1, X, …, X^D, 1, X`, seed count D+3) as the Shoup seed; +its length and degree lemmas apply verbatim. -/ + +/-- The SRS-seeded Shoup initial state: same seed table as the Maurer/all-pairs files, empty +pattern-history. -/ +noncomputable def srsStShoup (D : ℕ) : ShSt p := ⟨(srsSt (p := p) D).table, []⟩ + +@[simp] lemma srsStShoup_table (D : ℕ) : (srsStShoup (p := p) D).table = (srsSt (p := p) D).table := + rfl + +/-- **THE RANDOM-ENCODING (SHOUP) GGM t-SDH BOUND (sorry-free; Tier 1).** +Every free-comparison generic strategy — one that observes, at each step and FOR FREE, the full +equality pattern of all its held encodings — wins t-SDH against the real encoding oracle on at +most a `(C(fuel+D+4, 2)·D + (D+1))/(p−1)` fraction of trapdoors: the all-pairs collision event +(now TIGHT, +because comparison is free) plus the static Boneh–Boyen root event. SAME NUMERATOR as +`GgmRandomEncoding.rand_encoding_bound_srs_D`; the difference is the MODEL in which it is proved +(random-encoding free-comparison, not Maurer explicit-equality). The degree invariants are +DISCHARGED here (not assumed), and the identical-until-bad hybrid `runShoup_congr_off_bad` is +PROVEN. -/ +theorem shoup_ggm_sound (strat : ShoupStrat p) (fuel D : ℕ) (hD : 1 ≤ D) (hp : 2 ≤ p) : + shoupExperiment strat (srsStShoup D) fuel + ≤ (((fuel + D + 4).choose 2 * D + (D + 1) : ℕ) : ℚ) / (p - 1) := by + unfold shoupExperiment + have hseed : ∀ q ∈ (srsStShoup (p := p) D).table, q.natDegree ≤ D := by + rw [srsStShoup_table]; exact srsSt_table_natDegree_le D hD + have hlen : (srsStShoup (p := p) D).table.length + fuel + 1 ≤ fuel + D + 4 := by + rw [srsStShoup_table, srsSt_table_length]; omega + have hcard := card_realWinSetShoup_le_encoding strat (srsStShoup D) fuel D (fuel + D + 4) + (runShoup_output_natDegree_le symAns strat fuel (srsStShoup D) hseed) + (handleSetShoup_natDegree_le strat (srsStShoup D) fuel hseed) hlen + have hnum : ((realWinSetShoup strat (srsStShoup D) fuel).card : ℚ) + ≤ (((fuel + D + 4).choose 2 * D + (D + 1) : ℕ) : ℚ) := by exact_mod_cast hcard + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by exact_mod_cast hp + linarith + gcongr + +omit [Fact (Nat.Prime p)] in +/-- **Non-vacuity of the Shoup bound.** Whenever `C(fuel+D+4, 2)·D + (D+1) < p − 1` the bound is a +genuine rational `< 1` — at cryptographic parameters (`p ≈ 2²⁵⁴`, `D ≈ 2²⁰`, `fuel ≈ 2⁶⁰`) it is +`≈ 2⁻¹¹⁴`. Same non-vacuity shape as `rand_encoding_bound_lt_one`. -/ +theorem shoup_ggm_sound_lt_one (fuel D : ℕ) + (hreg : (fuel + D + 4).choose 2 * D + (D + 1) < p - 1) (hp : 2 ≤ p) : + (((fuel + D + 4).choose 2 * D + (D + 1) : ℕ) : ℚ) / (p - 1) < 1 := by + have hden : (0 : ℚ) < (p : ℚ) - 1 := by + have : (2 : ℚ) ≤ (p : ℚ) := by exact_mod_cast hp + linarith + rw [div_lt_one hden] + have h1 : (((fuel + D + 4).choose 2 * D + (D + 1) : ℕ) : ℚ) < ((p - 1 : ℕ) : ℚ) := by + exact_mod_cast hreg + have h2 : ((p - 1 : ℕ) : ℚ) = (p : ℚ) - 1 := by + have : (1 : ℕ) ≤ p := by omega + push_cast [Nat.cast_sub this]; ring + rw [h2] at h1; exact h1 + +end GgmShoup + +-- Axiom receipts: every headline theorem is sorry-free on the standard three axioms. +#print axioms GgmShoup.runShoup_congr_off_bad +#print axioms GgmShoup.realWinSetShoup_subset +#print axioms GgmShoup.card_realWinSetShoup_le_allPairs +#print axioms GgmShoup.shoup_ggm_sound +#print axioms GgmShoup.shoup_ggm_sound_lt_one diff --git a/ArkLib/Scratch/KzgVacuity/GgmShoupEmbed.lean b/ArkLib/Scratch/KzgVacuity/GgmShoupEmbed.lean new file mode 100644 index 0000000000..ed92392024 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/GgmShoupEmbed.lean @@ -0,0 +1,439 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Scratch.KzgVacuity.GgmShoup +import ArkLib.Scratch.KzgVacuity.GgmEndToEnd + +/-! +# The random-encoding (Shoup) GGM capstone for ArkLib's t-SDH experiment + +This file wires the random-encoding (Shoup) free-comparison strategy model +(`GgmShoup.ShoupStrat` / `runShoup`) into ArkLib's real `tSdhAdversary` / +`tSdhExperiment`, exactly as the Maurer explicit-equality track is wired via +`GgmEmbed.embed`. With both tracks in place, the two standard GGM formulations +bound ArkLib's actual $t$-SDH experiment; the Shoup track is no longer +standalone. The headline theorem is `shoup_tSdh_ggm_sound` (with its genuine +`< 1` corollary `shoup_tSdh_ggm_sound_lt_one`): for every free-comparison +strategy it bounds the winning probability of the embedded adversary +`embedShoup strat` against the real experiment, composed through one socket +exactly as the Maurer capstone `tSdh_ggm_sound`, via an `embedShoup` into +ArkLib's adversary type. + +## Faithfulness — free comparison realized in the concrete group + +The strategy model is the Shoup random-encoding one [Sho97], in which equality +of handles is observed for free. A real `tSdhAdversary` holds actual $G_1$ +elements and can test equality of any two it holds for free (`DecidableEq G₁`, +classically). In a prime-order group the exponent encoding $a \mapsto +g_1^{a.\mathrm{val}}$ is injective (`GgmArkLibTransport.gpow_val_inj_iff`), so +real group equality of two realized handles +$g_1^{(f\,\tau).\mathrm{val}} =? g_1^{(h\,\tau).\mathrm{val}}$ equals eval-at-$\tau$ +equality $f.\mathrm{eval}\,\tau =? h.\mathrm{eval}\,\tau$, i.e. +`GgmAdaptive.realAns τ`. Therefore the full pairwise-equality matrix of the +adversary's realized handles equals the symbolic `GgmShoup.eqPattern (realAns τ)` +— the free-comparison observation the strategy branches on. The injective +encoding genuinely realizes `eqPattern` off the bad event; free comparison is +not assumed, it is discharged by injectivity (`groupEqPattern_eq`). + +## The construction (matrix-valued analogue of `GgmEmbed`) + +* `groupEqPattern tableG` — the $|tableG| \times |tableG|$ real-group equality + matrix (the free-comparison observation a `tSdhAdversary` computes at zero cost + via `DecidableEq G₁`). +* `runEmbedAuxShoup` / `runEmbedShoup` / `embedShoup` — mirror `runEmbedAux` / + `runEmbed` / `embed` but thread the matrix pattern-history + (`GgmShoup.ShoupStrat`'s input), interpreting `lin` by real group products + (`GgmEmbed.combineG`) and observing `groupEqPattern` (no `query` move — + equality is ambient, as in the Shoup model). +* `groupEqPattern_eq` (the crux, one dimension richer than `GgmEmbed`'s + single-bool `hans`) — under the table↔polynomial invariant `IsEncoding`, the + group matrix equals `eqPattern (realAns τ)`, entrywise via `gpow_val_inj_iff`. +* `embedShoup_run_correspondence` — `runEmbedShoup` on the real SRS reproduces + `runShoup (realAns τ)` realized in the group ($g_1^{(output\,\tau).\mathrm{val}}$). + Mirrors `GgmEmbed.embed_run_correspondence`, reusing `isEncoding_append` / + `seedG_isEncoding`. + +## The capstone + +`shoup_tSdh_ggm_sound` composes exactly as the Maurer capstone `tSdh_ggm_sound`, +whose explicit-equality strategy model is Maurer's [Mau05]: + +* `Ggm.ProbThreading.experiment_eq_count` (adversary-agnostic) collapses ArkLib's + `tSdhExperiment` to a $\mathrm{Fin}\,(p-1)$ count, its determinism discharged by + `embedShoup_det`; +* `winIndexShoup_card_le` (reindex $i \mapsto i+1$, transported by + `GgmArkLibTransport.tSdhCondition_iff_field`) bounds that count by + `GgmShoup.realWinSetShoup.card`; +* `GgmShoup.card_realWinSetShoup_le_encoding` (the all-pairs Schwartz–Zippel + bound) bounds that by $\binom{fuel+D+4}{2} \cdot D + (D+1)$. + +The RHS is byte-identical to `tSdh_ggm_sound`'s: both embedded classes realize +the same $g_1^{(f\,\tau)}$ outputs and the same $t$-SDH win set (the Boneh–Boyen +$t$-SDH problem [BB04]); the two models differ only in how comparison is priced. +The bound quantifies over the image of `embedShoup`, not the full `tSdhAdversary` +type (over which it is false). + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +* [Maurer, U., *Abstract Models of Computation in Cryptography*][Mau05] +-/ + +open Polynomial Groups OracleSpec OracleComp +open scoped Classical NNReal ENNReal + +namespace GgmShoupEmbed + +open GgmCandidate GgmAdaptive GgmRandomEncoding GgmArkLibTransport GgmEmbed Ggm.ProbThreading +open GgmDegreeDischarge GgmShoup GgmEndToEnd + +variable {p : ℕ} [Fact (Nat.Prime p)] + [∀ i, SampleableType (unifSpec.Range i)] + {G₁ : Type} [Group G₁] [PrimeOrderWith G₁ p] {g₁ : G₁} + {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] {g₂ : G₂} + {D : ℕ} + +/-! ## 0. Sigma-matrix extensionality — two equality matrices over lists of equal length agree. -/ + +omit [∀ i, SampleableType (unifSpec.Range i)] in +/-- Two `Σ n, Fin n → Fin n → Bool`-packaged matrices are equal when their dimensions are equal and +the matrices agree entrywise (through the dimension cast). The dimension mismatch is discharged by +`subst`, so no `HEq`/`Fin` bookkeeping leaks into the callers. -/ +lemma sigma_matrix_ext {n m : ℕ} (h : n = m) + {f : Fin n → Fin n → Bool} {g' : Fin m → Fin m → Bool} + (hfg : ∀ (i j : Fin n), f i j = g' (Fin.cast h i) (Fin.cast h j)) : + (⟨n, f⟩ : Σ k : ℕ, Fin k → Fin k → Bool) = ⟨m, g'⟩ := by + subst h + refine Sigma.ext rfl (heq_of_eq (funext fun i => funext fun j => ?_)) + simpa using hfg i j + +/-! ## 1. The free-comparison observation in the real group, and its agreement with `eqPattern`. + +`groupEqPattern` is the `|tableG|×|tableG|` real-group equality matrix — exactly what a concrete +`tSdhAdversary` observes for free via `DecidableEq G₁`. Under `GgmEmbed.IsEncoding`, injectivity of +the +encoding folds it onto the symbolic `eqPattern (realAns τ)`. This is the free-comparison analogue of +`GgmEmbed.runEmbedAux_correspondence`'s single-bool `hans`, one dimension richer. -/ + +/-- The real-group free-comparison observation: the full pairwise `groupEq` matrix of the handle +table, packaged with its dimension (mirrors `GgmShoup.eqPattern`, over group elements). -/ +noncomputable def groupEqPattern (tableG : List G₁) : Σ n : ℕ, Fin n → Fin n → Bool := + ⟨tableG.length, fun i j => groupEq (tableG.get i) (tableG.get j)⟩ + +/-- **The free-comparison realization is FAITHFUL.** Under the table↔polynomial invariant, the +real-group equality matrix `groupEqPattern tableG` equals the symbolic eval-at-τ matrix +`eqPattern (realAns τ) table`: entrywise, real group equality of realized handles +`g^(fᵢ τ).val =? g^(fⱼ τ).val` folds to `fᵢ τ =? fⱼ τ` by injectivity (`gpow_val_inj_iff`). -/ +lemma groupEqPattern_eq {g : G₁} (hord : orderOf g = p) {τ : ZMod p} + {tableG : List G₁} {table : List ((ZMod p)[X])} (hInv : IsEncoding g τ tableG table) : + groupEqPattern tableG = eqPattern (realAns τ) table := by + obtain ⟨hlen, hpt⟩ := hInv + unfold groupEqPattern eqPattern + refine sigma_matrix_ext hlen ?_ + intro i j + have key : ∀ k : Fin tableG.length, + tableG.get k = g ^ ((table.get (Fin.cast hlen k)).eval τ).val := by + intro k + have hk := hpt (k : ℕ) + rw [List.getD_eq_getElem tableG 1 k.isLt, List.getD_eq_getElem table 0 (hlen ▸ k.isLt)] at hk + rw [List.get_eq_getElem, List.get_eq_getElem] + simpa using hk + simp only [groupEq, realAns] + exact decide_eq_decide.mpr (by rw [key i, key j]; exact gpow_val_inj_iff hord) + +/-! ## 2. `runEmbedShoup` — the matrix-threaded real-group run — and `embedShoup`. -/ + +/-- **The real-group Shoup run.** A `List G₁` handle table (no polynomials, no τ), evolved by +`strat`'s `lin` moves interpreted as REAL group products (`GgmEmbed.combineG`, REUSED). At every +step +it observes the FULL real-group equality matrix `groupEqPattern` for free — the random-encoding +free-comparison discipline — and appends it to the pattern-history the strategy branches on. There +is +no `query` branch (equality is ambient). Mirrors `GgmEmbed.runEmbedAux`, matrix-valued. -/ +noncomputable def runEmbedAuxShoup (g : G₁) (strat : ShoupStrat p) : + ℕ → (List G₁ × List (Σ n : ℕ, Fin n → Fin n → Bool)) → Option (ZMod p × G₁) + | 0, _ => some (0, 1) + | fuel + 1, (tableG, phist) => + match strat (phist ++ [groupEqPattern tableG]) with + | Sum.inr (c, k) => some (c, tableG.getD k 1) + | Sum.inl (ShoupMove.lin spec) => + runEmbedAuxShoup g strat fuel + (tableG ++ [combineG spec tableG], phist ++ [groupEqPattern tableG]) + +/-- **`runEmbedShoup`** — run the free-comparison strategy against the real-group SRS. Reads only +the +G₁ tower (`srs.1`); being pairing-free it needs neither the G₂ generator nor τ. Mirrors +`GgmEmbed.runEmbed`. -/ +noncomputable def runEmbedShoup (g₁ : G₁) (D fuel : ℕ) (strat : ShoupStrat p) + (srs : Vector G₁ (D + 1) × Vector G₂ 2) : Option (ZMod p × G₁) := + runEmbedAuxShoup g₁ strat fuel (seedG srs.1.toList D, []) + +/-- **`embedShoup : ShoupStrat p → tSdhAdversary D`** — the concrete adversary that realizes free +comparison. It holds real `G₁` elements and, at each step, computes the full `groupEqPattern` of +them +(free `DecidableEq G₁`), feeding the strategy the REAL equality matrix — which off the bad event IS +the symbolic `eqPattern (realAns τ)` (`groupEqPattern_eq`). Deterministic, empty-cache; its IMAGE is +the free-comparison adversary class `shoup_tSdh_ggm_sound` quantifies over. Mirrors +`GgmEmbed.embed`. -/ +noncomputable def embedShoup (g₁ : G₁) (D fuel : ℕ) (strat : ShoupStrat p) : + Groups.tSdhAdversary D (G₁ := G₁) (G₂ := G₂) (p := p) := + fun srs => pure (runEmbedShoup g₁ D fuel strat srs) + +/-! ## 3. The correspondence: `runEmbedShoup` steps in lockstep with `runShoup (realAns τ)`. -/ + +/-- **THE CORRESPONDENCE (induction core).** Under the invariant, `runEmbedAuxShoup` on the group +table returns exactly the committed offset of `runShoup (realAns τ)` and the real-group encoding of +its committed output polynomial. The two runs step in lockstep because at each step the group's FREE +equality matrix equals the symbolic one (`groupEqPattern_eq`), so the strategy makes the same +decision. Mirrors `GgmEmbed.runEmbedAux_correspondence`, matrix-valued (and simpler — Shoup has no +query log to thread). -/ +lemma runEmbedAuxShoup_correspondence {g : G₁} (hord : orderOf g = p) (τ : ZMod p) + (strat : ShoupStrat p) : + ∀ (fuel : ℕ) (tableG : List G₁) (table : List ((ZMod p)[X])) + (phist : List (Σ n : ℕ, Fin n → Fin n → Bool)), + IsEncoding g τ tableG table → + runEmbedAuxShoup g strat fuel (tableG, phist) + = some ((runShoup (realAns τ) strat fuel ⟨table, phist⟩).1, + g ^ ((runShoup (realAns τ) strat fuel ⟨table, phist⟩).2.eval τ).val) := by + intro fuel + induction fuel with + | zero => + intro tableG table phist hInv + simp only [runEmbedAuxShoup, runShoup, eval_zero, encode_zero] + | succ fuel ih => + intro tableG table phist hInv + have hpat : groupEqPattern tableG = eqPattern (realAns τ) table := groupEqPattern_eq hord hInv + rcases hdec : strat (phist ++ [eqPattern (realAns τ) table]) with m | out + · cases m with + | lin spec => + have eG : runEmbedAuxShoup g strat (fuel + 1) (tableG, phist) + = runEmbedAuxShoup g strat fuel + (tableG ++ [combineG spec tableG], phist ++ [eqPattern (realAns τ) table]) := by + simp only [runEmbedAuxShoup, hpat, hdec] + have eS : runShoup (realAns τ) strat (fuel + 1) ⟨table, phist⟩ + = runShoup (realAns τ) strat fuel + ⟨table ++ [combine spec table], phist ++ [eqPattern (realAns τ) table]⟩ := by + simp only [runShoup, hdec] + rw [eG, eS] + exact ih _ _ _ (isEncoding_append hord spec hInv) + · have eG : runEmbedAuxShoup g strat (fuel + 1) (tableG, phist) + = some (out.1, tableG.getD out.2 1) := by + simp only [runEmbedAuxShoup, hpat, hdec] + have eS : runShoup (realAns τ) strat (fuel + 1) ⟨table, phist⟩ + = (out.1, table.getD out.2 0) := by + simp only [runShoup, hdec] + rw [eG, eS, hInv.2 out.2] + +omit [∀ i, SampleableType (unifSpec.Range i)] [PrimeOrderWith G₂ p] in +/-- **THE DELIVERABLE.** `runEmbedShoup` on the real SRS `PowerSrs.generate D τ` returns the +committed +offset of the SYMBOLIC free-comparison run `runShoup (realAns τ) strat fuel (srsStShoup D)` and the +real-group encoding `g₁ ^ (output.eval τ).val` of its committed output polynomial. This certifies +`embedShoup strat` is genuinely generic (it reproduces the symbolic Shoup run realized in the group, +never inverting the encoding), and is the socket the capstone consumes. Mirrors +`GgmEmbed.embed_run_correspondence`. -/ +theorem embedShoup_run_correspondence (hord : orderOf g₁ = p) + (D : ℕ) (hD : 1 ≤ D) (τ : ZMod p) (strat : ShoupStrat p) (fuel : ℕ) : + runEmbedShoup g₁ D fuel strat (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ) + = some ((runShoup (realAns τ) strat fuel (srsStShoup D)).1, + g₁ ^ ((runShoup (realAns τ) strat fuel (srsStShoup D)).2.eval τ).val) := by + have hsrs1 : (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ).1 = PowerSrs.tower g₁ τ D := rfl + rw [runEmbedShoup, hsrs1, + runEmbedAuxShoup_correspondence hord τ strat fuel + (seedG (PowerSrs.tower g₁ τ D).toList D) (srsSt (p := p) D).table [] + (seedG_isEncoding g₁ hord τ D hD)] + rfl + +/-! ## 4. The deterministic output of `embedShoup`, and `embedShoup`'s determinism. -/ + +/-- The deterministic-given-τ `Option`-output of `embedShoup strat` on the SRS generated from `τ`: +the committed offset of the symbolic Shoup run realized in the group. This is the RHS of +`embedShoup_run_correspondence`, packaged as the `resultOf` that `experiment_eq_count` consumes. +Mirrors `GgmEndToEnd.stratResult`. -/ +noncomputable def stratResultShoup (g₁ : G₁) (D fuel : ℕ) (strat : ShoupStrat p) : + ZMod p → Option (ZMod p × G₁) := + fun τ => some ((runShoup (realAns τ) strat fuel (srsStShoup D)).1, + g₁ ^ ((runShoup (realAns τ) strat fuel (srsStShoup D)).2.eval τ).val) + +omit [∀ i, SampleableType (unifSpec.Range i)] [PrimeOrderWith G₂ p] in +/-- **`embedShoup strat` is deterministic-given-τ from the empty cache**, with output +`stratResultShoup`. `embedShoup strat srs = pure (runEmbedShoup … srs)`, and +`embedShoup_run_correspondence` identifies `runEmbedShoup …` on `PowerSrs.generate D τ` with +`stratResultShoup τ`. This is the exact `hdet` hypothesis of `experiment_eq_count`. Mirrors +`GgmEndToEnd.embed_det`. -/ +theorem embedShoup_det (hord₁ : orderOf g₁ = p) (hD : 1 ≤ D) (strat : ShoupStrat p) (fuel : ℕ) : + ∀ τ, (embedShoup g₁ D fuel strat + (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ)).run' ∅ + = pure (stratResultShoup g₁ D fuel strat τ) := by + intro τ + simp only [embedShoup, StateT.run'_pure', stratResultShoup] + rw [embedShoup_run_correspondence hord₁ D hD τ strat fuel] + +/-! ## 5. The reindex `Fin (p−1) ↪ realWinSetShoup`: winning-index count ≤ winning-set card. -/ + +omit [∀ i, SampleableType (unifSpec.Range i)] in +/-- **The reindex bound.** The number of winning nonzero-trapdoor indices `i : Fin (p−1)` (those on +which `embedShoup strat`'s deterministic output `stratResultShoup` wins ArkLib's `tSdhCondition`) is +at most the cardinality of the field-level free-comparison winning set `realWinSetShoup`. Proved by +the injection `i ↦ (i+1 : ZMod p)` into `realWinSetShoup`, transporting each winning index through +`tSdhCondition_iff_field`. Mirrors `GgmEndToEnd.winIndex_card_le`, targeting `realWinSetShoup`. -/ +theorem winIndexShoup_card_le (hord₁ : orderOf g₁ = p) (hp : 2 ≤ p) + (strat : ShoupStrat p) (fuel : ℕ) : + (Finset.univ.filter (winPred (stratResultShoup g₁ D fuel strat) g₁)).card + ≤ (realWinSetShoup strat (srsStShoup D) fuel).card := by + refine Finset.card_le_card_of_injOn (fun i => ((i : ℕ) + 1 : ZMod p)) ?_ ?_ + · -- maps winning indices into `realWinSetShoup` + intro i hi + rw [Finset.mem_coe, Finset.mem_filter] at hi + have hw := hi.2 + simp only [winPred] at hw + obtain ⟨ch, hres, hcond⟩ := hw + rw [stratResultShoup, Option.some.injEq] at hres + rw [← hres] at hcond + rw [Finset.mem_coe, realWinSetShoup, Finset.mem_filter] + refine ⟨?_, (tSdhCondition_iff_field hord₁ _ _ _).mp hcond⟩ + rw [nonzeroPoints, Finset.mem_erase] + exact ⟨index_ne_zero hp i, Finset.mem_univ _⟩ + · -- `i ↦ (i+1 : ZMod p)` is injective on `Fin (p−1)` + intro i _ j _ hij + simp only at hij + have hi : (i : ℕ) + 1 < p := by have := i.isLt; omega + have hj : (j : ℕ) + 1 < p := by have := j.isLt; omega + have hci : ((i : ℕ) + 1 : ZMod p) = (((i : ℕ) + 1 : ℕ) : ZMod p) := by push_cast; ring + have hcj : ((j : ℕ) + 1 : ZMod p) = (((j : ℕ) + 1 : ℕ) : ZMod p) := by push_cast; ring + rw [hci, hcj] at hij + have := congrArg ZMod.val hij + rw [ZMod.val_natCast_of_lt hi, ZMod.val_natCast_of_lt hj] at this + exact Fin.ext (by omega) + +/-! ## 6. ⚑ THE CAPSTONE — the Shoup track wired to ArkLib's REAL `tSdhExperiment`. -/ + +omit [PrimeOrderWith G₂ p] in +/-- **`shoup_tSdh_ggm_sound` — the random-encoding (Shoup) GGM t-SDH bound about ArkLib's REAL +experiment.** For every free-comparison strategy `strat : ShoupStrat p`, the embedded ArkLib +adversary +`embedShoup strat` wins ArkLib's REAL t-SDH experiment with probability at most +`(C(fuel+D+4, 2)·D + (D+1)) / (p − 1)` — the all-pairs collision number (TIGHT under free +comparison) +plus the static Boneh–Boyen root event. Composed through ONE socket, EXACTLY as the Maurer capstone +`GgmEndToEnd.tSdh_ggm_sound`: + +* `experiment_eq_count` (adversary-agnostic) turns `tSdhExperiment` into a `Fin (p−1)` count, its + determinism discharged by `embedShoup_det` (resting on `embedShoup_run_correspondence`); +* the count is bounded by `realWinSetShoup.card` via `winIndexShoup_card_le` + + `tSdhCondition_iff_field`; +* `realWinSetShoup.card` is bounded by `GgmShoup.card_realWinSetShoup_le_encoding` (the all-pairs + Schwartz–Zippel bound, whose hybrid `runShoup_congr_off_bad` is PROVEN in Tier 1), with both + degree + invariants discharged. + +RHS byte-identical to `tSdh_ggm_sound`: BOTH standard GGM models now bound the SAME real experiment. +Quantifies over the IMAGE of `embedShoup` — NOT the full `tSdhAdversary` type (over which it is +false). -/ +theorem shoup_tSdh_ggm_sound + (hord₁ : orderOf g₁ = p) + (hD : 1 ≤ D) (strat : ShoupStrat p) (fuel : ℕ) : + tSdhExperiment (g₁ := g₁) (g₂ := g₂) D (embedShoup g₁ D fuel strat) + ≤ (((fuel + D + 4).choose 2 * D + (D + 1) : ℕ) : ℝ≥0∞) / ((p - 1 : ℕ) : ℝ≥0∞) := by + have hp : 2 ≤ p := (Fact.out : Nat.Prime p).two_le + have hseed : ∀ q ∈ (srsStShoup (p := p) D).table, q.natDegree ≤ D := by + rw [srsStShoup_table]; exact srsSt_table_natDegree_le D hD + -- (C) collapse the experiment to a count over `Fin (p−1)` + rw [experiment_eq_count D (embedShoup g₁ D fuel strat) (stratResultShoup g₁ D fuel strat) + (embedShoup_det hord₁ hD strat fuel)] + -- the numerator is bounded, in ℕ, by the Shoup all-pairs number + have hcard : (Finset.univ.filter (winPred (stratResultShoup g₁ D fuel strat) g₁)).card + ≤ (fuel + D + 4).choose 2 * D + (D + 1) := by + refine (winIndexShoup_card_le hord₁ hp strat fuel).trans ?_ + exact card_realWinSetShoup_le_encoding strat (srsStShoup D) fuel D (fuel + D + 4) + (runShoup_output_natDegree_le symAns strat fuel (srsStShoup D) hseed) + (handleSetShoup_natDegree_le strat (srsStShoup D) fuel hseed) + (by rw [srsStShoup_table, srsSt_table_length]; omega) + -- lift the ℕ count bound through the ℝ≥0∞ division + exact ENNReal.div_le_div_right (by exact_mod_cast hcard) _ + +omit [PrimeOrderWith G₂ p] in +/-- **`shoup_tSdh_ggm_sound_lt_one`.** Under the standard security regime +`C(fuel+D+4, 2)·D + (D+1) < p − 1`, the t-SDH advantage of `embedShoup strat` against ArkLib's REAL +experiment is a genuine `< 1` — real content, not a restated `≤ 1`. Mirrors +`GgmEndToEnd.tSdh_ggm_sound_lt_one`. -/ +theorem shoup_tSdh_ggm_sound_lt_one + (hord₁ : orderOf g₁ = p) + (hD : 1 ≤ D) (strat : ShoupStrat p) (fuel : ℕ) + (hreg : (fuel + D + 4).choose 2 * D + (D + 1) < p - 1) : + tSdhExperiment (g₁ := g₁) (g₂ := g₂) D (embedShoup g₁ D fuel strat) < 1 := by + refine lt_of_le_of_lt (shoup_tSdh_ggm_sound hord₁ hD strat fuel) ?_ + have hb0 : ((p - 1 : ℕ) : ℝ≥0∞) ≠ 0 := by + rw [Ne, Nat.cast_eq_zero]; omega + have hbtop : ((p - 1 : ℕ) : ℝ≥0∞) ≠ ⊤ := ENNReal.natCast_ne_top _ + rw [ENNReal.div_lt_iff (Or.inl hb0) (Or.inl hbtop), one_mul] + exact_mod_cast hreg + +/-! ## 7. Non-collapse: the embedded free-comparison class is a genuine non-singleton. + +`shoup_tSdh_ggm_sound` quantifies over the IMAGE of `embedShoup`, NOT over all `tSdhAdversary` (over +which the bound is FALSE — a `Classical.choice`-definable adversary inverts the encoding and wins +with +probability 1). `embedShoup_noncollapsing` certifies distinct strategies give distinct real +adversaries, so the class is not degenerate. Mirrors `GgmEmbed.embed_noncollapsing`. -/ + +/-- The constant Shoup strategy: immediately commit offset `c`, read handle `0`, ignoring the +pattern +history. `stratOffsetShoup 0` / `stratOffsetShoup 1` are the non-collapse witnesses. -/ +private def stratOffsetShoup (c : ZMod p) : ShoupStrat p := fun _ => Sum.inr (c, 0) + +omit [∀ i, SampleableType (unifSpec.Range i)] [Group G₂] [PrimeOrderWith G₂ p] in +/-- A `stratOffsetShoup c` run (≥ 1 fuel) commits `c` and the `0`-th seed handle — the SRS's `0`-th +G₁ element — on ANY SRS. Reuses `GgmEmbed.seedG_getD_zero`. -/ +lemma runEmbedShoup_stratOffsetShoup (D f : ℕ) (c : ZMod p) + (srs : Vector G₁ (D + 1) × Vector G₂ 2) : + runEmbedShoup g₁ D (f + 1) (stratOffsetShoup c) srs + = some (c, srs.1.toList.getD 0 1) := by + simp only [runEmbedShoup, runEmbedAuxShoup, stratOffsetShoup, seedG_getD_zero] + +omit [∀ i, SampleableType (unifSpec.Range i)] [PrimeOrderWith G₂ p] in +/-- **`embedShoup_noncollapsing` — distinct strategies → distinct real adversaries.** There are two +free-comparison strategies whose `embedShoup`-outputs (deterministic `runEmbedShoup` values) are +distinct, so the IMAGE of `embedShoup` is a genuine non-singleton: + +* (i) On EVERY SRS, `stratOffsetShoup 0` and `stratOffsetShoup 1` commit distinct offsets (`0 ≠ 1`). +* (ii) On the real KZG SRS, `stratOffsetShoup 0`'s committed GROUP element is the base generator + `g₁` + (`≠ 1` by `hg₁`), so the image really exercises the group. + +Honest non-claim: `embedShoup` is not injective (off-branch pattern disagreement is invisible); this +asserts only that the image is not a singleton — what a meaningful quantifier requires. -/ +theorem embedShoup_noncollapsing (hg₁ : g₁ ≠ 1) (D f : ℕ) : + ∃ s₀ s₁ : ShoupStrat p, + (∀ srs : Vector G₁ (D + 1) × Vector G₂ 2, + runEmbedShoup g₁ D (f + 1) s₀ srs ≠ runEmbedShoup g₁ D (f + 1) s₁ srs) ∧ + (∀ τ : ZMod p, + runEmbedShoup g₁ D (f + 1) s₀ (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ) + ≠ some (0, 1)) := by + have h01 : (0 : ZMod p) ≠ 1 := zero_ne_one + refine ⟨stratOffsetShoup 0, stratOffsetShoup 1, ?_, ?_⟩ + · intro srs h + rw [runEmbedShoup_stratOffsetShoup D f 0 srs, runEmbedShoup_stratOffsetShoup D f 1 srs, + Option.some.injEq, Prod.mk.injEq] at h + exact h01 h.1 + · intro τ h + have hval : (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ).1.toList.getD 0 1 = g₁ := by + have hsrs1 : (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ).1 = PowerSrs.tower g₁ τ D := rfl + rw [hsrs1, tower_toList_getD τ D 0 (Nat.succ_pos D), pow_zero, pow_one] + rw [runEmbedShoup_stratOffsetShoup D f 0 (PowerSrs.generate (g₁ := g₁) (g₂ := g₂) D τ), hval, + Option.some.injEq, Prod.mk.injEq] at h + exact hg₁ h.2 + +/-! ## Axiom hygiene — the wiring rests on exactly `[propext, Classical.choice, Quot.sound]`. -/ + +#print axioms embedShoup_run_correspondence +#print axioms embedShoup_det +#print axioms winIndexShoup_card_le +#print axioms shoup_tSdh_ggm_sound +#print axioms shoup_tSdh_ggm_sound_lt_one +#print axioms embedShoup_noncollapsing + +end GgmShoupEmbed diff --git a/ArkLib/Scratch/KzgVacuity/KzgVacuity.lean b/ArkLib/Scratch/KzgVacuity/KzgVacuity.lean new file mode 100644 index 0000000000..8b94fc796d --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/KzgVacuity.lean @@ -0,0 +1,377 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Commitments.Functional.KZG.Binding +import ArkLib.Commitments.Functional.KZG.FunctionBinding.Support + +/-! +# Vacuity of the concrete-group `tSdhAssumption` and `arsdhAssumption` + +This file mechanizes a refutation of ArkLib's `Groups.tSdhAssumption` and `Groups.arsdhAssumption` +as stated over *concrete* prime-order groups. The $t$-SDH assumption [BB04] underlies the KZG +polynomial commitment scheme [KZG10]; ArkLib's hardness game hands its adversary the structured +reference string as concrete group elements, including the verifier leg $g_2^{\tau}$. + +The refutation is that a `Classical.choice`-definable adversary is a legal inhabitant of the +adversary type. From $g_2^{\tau}$ the extractor `dlogOf` (built from ArkLib's own +`Groups.exists_zmod_power_of_generator`) recovers the trapdoor $\tau : \mathbb{Z}/p$ and returns a +winning element with probability $1$: + +* `not_tSdhAssumption` — `tauExtractingAdversary` wins `tSdhExperiment` below every error `< 1`. +* `binding_hypotheses_unsatisfiable` — the hypotheses bundling KZG binding to `tSdhAssumption` + are jointly unsatisfiable in the concrete-group model. +* `not_arsdhAssumption` / `arsdh_binding_hypotheses_unsatisfiable` — the same for the algebraic + RSDH variant, using `arsdhExtractingAdversary` on a challenge set avoiding the trapdoor. + +The finding is that hardness must be quantified over a *restricted* adversary class — the generic +group model [Sho97], [Mau05], or the algebraic group model [FKL18] — where the trapdoor is never +in the adversary's view. The companion `GgmEndToEnd` file supplies that generic bound. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Kate, A., Zaverucha, G. M., and Goldberg, I., *Constant-Size Commitments to Polynomials and + Their Applications*][KZG10] +* [Shoup, V., *Lower Bounds for Discrete Logarithms and Related Problems*][Sho97] +* [Maurer, U., *Abstract Models of Computation in Cryptography*][Mau05] +* [Fuchsbauer, G., Kiltz, E., and Loss, J., *The Algebraic Group Model and its Applications*][FKL18] +-/ + +open OracleSpec OracleComp +open scoped NNReal ENNReal + +namespace ArkLibVacuity + +section Dlog + +variable {p : ℕ} [Fact (Nat.Prime p)] + +/-- The choice-definable discrete logarithm base a nontrivial `g` in a prime-order group. +This is *not* an algorithm: it is `Exists.choose` applied to ArkLib's own +`Groups.exists_zmod_power_of_generator`. It is nevertheless a perfectly legal +inhabitant of `ZMod p`, and that is the whole point. -/ +noncomputable def dlogOf {G : Type} [Group G] [PrimeOrderWith G p] {g : G} (hg : g ≠ 1) + (x : G) : ZMod p := + (Groups.exists_zmod_power_of_generator (G := G) PrimeOrderWith.hCard hg + (Groups.orderOf_eq_prime_of_ne_one g hg) x).choose + +/-- `dlogOf` inverts exponentiation base a nontrivial element of a prime-order group. -/ +lemma dlogOf_pow {G : Type} [Group G] [PrimeOrderWith G p] {g : G} (hg : g ≠ 1) (a : ZMod p) : + dlogOf (p := p) hg (g ^ a.val) = a := by + have hord : orderOf g = p := Groups.orderOf_eq_prime_of_ne_one g hg + have hspec : g ^ a.val = g ^ (dlogOf (p := p) hg (g ^ a.val)).val := + (Groups.exists_zmod_power_of_generator (G := G) PrimeOrderWith.hCard hg hord + (g ^ a.val)).choose_spec + have hdiv : g ^ (dlogOf (p := p) hg (g ^ a.val) - a).val = 1 := by + rw [← Groups.gpow_div_eq hord _ a, ← hspec, div_self'] + exact sub_eq_zero.mp (Groups.zmod_eq_zero_of_gpow_eq_one hord hdiv) + +/-- Every value in the support of ArkLib's trapdoor sampler is nonzero. -/ +lemma sampleNonzeroZMod_ne_zero {τ : ZMod p} + (hτ : τ ∈ support (Groups.sampleNonzeroZMod (p := p))) : τ ≠ 0 := by + have hp : 1 < p := Nat.Prime.one_lt Fact.out + haveI : NeZero (p - 1) := ⟨Nat.pos_iff_ne_zero.mp (Nat.sub_pos_of_lt hp)⟩ + haveI : NeZero p := ⟨Nat.pos_iff_ne_zero.mp (Nat.zero_lt_of_lt hp)⟩ + rw [Groups.sampleNonzeroZMod, support_map] at hτ + obtain ⟨i, -, rfl⟩ := hτ + have hi := i.isLt + have hlt : (i : ℕ) + 1 < p := by omega + intro hzero + simp only at hzero + have hdvd : (((i : ℕ) + 1 : ℕ) : ZMod p) = 0 := by push_cast; exact hzero + rw [ZMod.natCast_eq_zero_iff] at hdvd + exact absurd (Nat.le_of_dvd (Nat.succ_pos _) hdvd) (not_le.mpr hlt) + +/-- ArkLib's trapdoor sampler never fails. -/ +lemma probFailure_sampleNonzeroZMod : Pr[⊥ | Groups.sampleNonzeroZMod (p := p)] = 0 := by + rw [Groups.sampleNonzeroZMod]; simp + +end Dlog + +section Refutation + +-- `PrimeOrderWith G₁ p` is deliberately absent: the t-SDH solution the adversary returns +-- lives in `G₁` as a bare group element, so nothing in this section needs `G₁` prime-order. +variable {p : ℕ} [Fact (Nat.Prime p)] + {G₁ : Type} [Group G₁] {g₁ : G₁} + {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] {g₂ : G₂} + [∀ i, SampleableType (unifSpec.Range i)] + +/-- The winning t-SDH adversary. It reads `g₂ ^ τ` out of the *verifier* leg of the SRS, +recovers `τ` by `Classical.choice`, and returns the t-SDH solution at offset `c = 0`. +It makes ZERO oracle queries: all of its work happens under `pure`, which the free monad +`ProbComp` does not charge for. -/ +noncomputable def tauExtractingAdversary (hg₂ : g₂ ≠ 1) (D : ℕ) : + Groups.tSdhAdversary (G₁ := G₁) (G₂ := G₂) (p := p) D := + fun srs => pure (some (0, g₁ ^ (1 / dlogOf (p := p) hg₂ srs.2[1]).val)) + +/-- The t-SDH game with the exhibited adversary collapses to a single `map` over the +trapdoor sampler: the adversary has already recovered `τ`. -/ +lemma game_run_eq (hg₂ : g₂ ≠ 1) (D : ℕ) : + (Groups.tSdhGame (g₁ := g₁) (g₂ := g₂) D + (tauExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D)).run + = (fun τ : ZMod p => some (τ, (0 : ZMod p), g₁ ^ (1 / τ).val)) + <$> Groups.sampleNonzeroZMod := by + simp [Groups.tSdhGame, tauExtractingAdversary, Groups.PowerSrs.generate, + Groups.PowerSrs.tower, dlogOf_pow hg₂] + +/-- The exhibited adversary wins the t-SDH game with probability exactly `1`. -/ +theorem tSdhExperiment_tauExtractingAdversary (hg₂ : g₂ ≠ 1) (D : ℕ) : + Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) D + (tauExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D) = 1 := by + classical + rw [Groups.tSdhExperiment, probEvent_eq_one_iff] + refine ⟨?_, ?_⟩ + · rw [OptionT.probFailure_eq, game_run_eq (g₁ := g₁) hg₂ D, probFailure_map, + probFailure_sampleNonzeroZMod] + simp + · intro x hx + rw [OptionT.support_def, game_run_eq (g₁ := g₁) hg₂ D, support_map] at hx + obtain ⟨τ, hτ, hxτ⟩ := hx + simp only [Option.some.injEq] at hxτ + subst hxτ + have hτ0 : τ ≠ 0 := sampleNonzeroZMod_ne_zero hτ + exact ⟨by simpa using hτ0, by simp⟩ + +/-- **The refutation.** ArkLib's `tSdhAssumption` is FALSE for every error bound `< 1`, +at every degree `D`, in every prime-order group pair with a nontrivial `g₂`. +No hypothesis about the size of `p` is needed: this is not an asymptotic statement. -/ +theorem not_tSdhAssumption (hg₂ : g₂ ≠ 1) (D : ℕ) (error : ℝ≥0) (herr : (error : ℝ≥0∞) < 1) : + ¬ Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) D error := by + intro h + have hle := h (tauExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D) + rw [tSdhExperiment_tauExtractingAdversary (g₁ := g₁) hg₂ D] at hle + exact absurd (lt_of_le_of_lt hle herr) (lt_irrefl 1) + +omit [PrimeOrderWith G₂ p] in +/-- **The other regime.** For any error bound `≥ 1`, `tSdhAssumption` holds *trivially*: a +success probability is always `≤ 1`. Combined with `not_tSdhAssumption` (false for `error < 1`), +this shows `tSdhAssumption` has NO content at ANY parameter — it is either false or vacuously +true. `probEvent_le_one` is the whole argument. -/ +theorem tSdhAssumption_trivial_of_one_le (D : ℕ) (error : ℝ≥0) + (herr : (1 : ℝ≥0∞) ≤ (error : ℝ≥0∞)) : + Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) D error := by + intro adversary + refine le_trans ?_ herr + rw [Groups.tSdhExperiment] + exact probEvent_le_one + +/-! ### Canary + +A gate that accepts everything is a broken gate. The two lemmas below check that +`tSdhExperiment` is not *constantly* `1` — i.e. that the probability-1 theorem above is a +statement about the exhibited adversary and not an artifact of the probability machinery. -/ + +/-- An adversary that simply gives up. -/ +def givingUpAdversary (D : ℕ) : Groups.tSdhAdversary (G₁ := G₁) (G₂ := G₂) (p := p) D := + fun _ => pure none + +omit [PrimeOrderWith G₂ p] in +/-- CANARY: giving up loses with probability `1`, so `tSdhExperiment` discriminates. -/ +theorem tSdhExperiment_givingUpAdversary (D : ℕ) : + Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) D + (givingUpAdversary (G₁ := G₁) (G₂ := G₂) (p := p) D) = 0 := by + classical + rw [Groups.tSdhExperiment, probEvent_eq_zero_iff] + intro x hx + rw [OptionT.support_def] at hx + simp [Groups.tSdhGame, givingUpAdversary] at hx + +/-- CANARY: consequently the probability-1 result is not vacuous — the two adversaries +are genuinely separated by the experiment. -/ +theorem experiment_discriminates (hg₂ : g₂ ≠ 1) (D : ℕ) : + Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) D + (givingUpAdversary (G₁ := G₁) (G₂ := G₂) (p := p) D) + ≠ Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) D + (tauExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D) := by + rw [tSdhExperiment_givingUpAdversary (g₁ := g₁) (g₂ := g₂) D, + tSdhExperiment_tauExtractingAdversary (g₁ := g₁) hg₂ D] + exact zero_ne_one + +end Refutation + +section BindingIsVacuous + +variable {p : ℕ} [Fact (Nat.Prime p)] + {G₁ : Type} [Group G₁] [PrimeOrderWith G₁ p] {g₁ : G₁} + {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] {g₂ : G₂} + {Gₜ : Type} [Group Gₜ] [PrimeOrderWith Gₜ p] + [Module (ZMod p) (Additive G₁)] [Module (ZMod p) (Additive G₂)] + [Module (ZMod p) (Additive Gₜ)] + [∀ i, SampleableType (unifSpec.Range i)] + +omit [∀ i, SampleableType (unifSpec.Range i)] in +/-- `binding`'s own pairing hypothesis forces the G₂ generator to be nontrivial, +because the pairing is `ZMod p`-bilinear and therefore kills the identity. -/ +lemma g₂_ne_one_of_pairing_ne_zero + (pairing : (Additive G₁) →ₗ[ZMod p] (Additive G₂) →ₗ[ZMod p] (Additive Gₜ)) + (hpair : pairing (Additive.ofMul g₁) (Additive.ofMul g₂) ≠ 0) : g₂ ≠ 1 := by + intro h + apply hpair + rw [show (Additive.ofMul g₂) = 0 from congrArg Additive.ofMul h] + exact map_zero _ + +/-- **`KZG.binding`'s hypotheses are jointly unsatisfiable at every meaningful error.** +The very pairing nondegeneracy that `binding` needs to run its reduction is what makes +its `t`-SDH premise false. So `binding` is only ever applicable with `tSdhError ≥ 1`, +where its conclusion is a triviality (a probability is always `≤ 1`). -/ +theorem binding_hypotheses_unsatisfiable + (pairing : (Additive G₁) →ₗ[ZMod p] (Additive G₂) →ₗ[ZMod p] (Additive Gₜ)) + (hpair : pairing (Additive.ofMul g₁) (Additive.ofMul g₂) ≠ 0) + (n : ℕ) (tSdhError : ℝ≥0) (herr : (tSdhError : ℝ≥0∞) < 1) : + ¬ Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) n tSdhError := + not_tSdhAssumption (g₁ := g₁) (g₂_ne_one_of_pairing_ne_zero pairing hpair) n tSdhError herr + +end BindingIsVacuous + +/-! ## ARSDH is vacuous in the `function_binding` parameter regime by the same argument + +ArkLib's `Groups.arsdhAssumption` (`Definition 9.6` in CGKY25, powering `KZG.function_binding`) +has the identical shape: `∀ adversary, arsdhExperiment D adversary ≤ error`, quantifying over the +adversary TYPE with no resource bound. In the parameter regime used by `function_binding`, it falls +the same two ways. The only extra work over the +`t`-SDH case is producing, for each trapdoor `τ`, a size-`D+1` set `S` with `τ ∉ S` (so the +vanishing polynomial `Z_S` does not vanish at `τ`); this requires `p ≥ D+2`, which is exactly the +`hp : p ≥ n + 2` hypothesis `function_binding` already carries. No claim is made here about the +separate degenerate regime `p < D + 2`, where a size-`D+1` set avoiding τ may not exist. -/ + +section ArsdhRefutation + +open CompPoly CompPoly.CPolynomial + +-- The combinatorial helpers below are group-free — they need only `ZMod p`. The group and +-- sampling instances enter with the second `variable` block, just before the adversary. +variable {p : ℕ} [Fact (Nat.Prime p)] + +/-- When `p ≥ D + 2` there is a size-`D+1` subset of `ZMod p` avoiding any given `τ`. +Not an algorithm — `Finset.exists_subset_card_eq` on `univ.erase τ`. -/ +lemma exists_finset_card_avoiding (D : ℕ) (hpD : D + 2 ≤ p) (τ : ZMod p) : + ∃ S : Finset (ZMod p), S.card = D + 1 ∧ τ ∉ S := by + haveI : NeZero p := ⟨Nat.pos_iff_ne_zero.mp (Nat.Prime.pos Fact.out)⟩ + have hcard : D + 1 ≤ (Finset.univ.erase τ).card := by + rw [Finset.card_erase_of_mem (Finset.mem_univ τ), Finset.card_univ, ZMod.card] + omega + obtain ⟨t, ht_sub, ht_card⟩ := Finset.exists_subset_card_eq hcard + exact ⟨t, ht_card, fun h => (Finset.mem_erase.mp (ht_sub h)).1 rfl⟩ + +/-- The trapdoor-indexed choice of avoiding set. Choice-definable, like `dlogOf`. -/ +noncomputable def chosenFinset (D : ℕ) (hpD : D + 2 ≤ p) (τ : ZMod p) : Finset (ZMod p) := + (exists_finset_card_avoiding (p := p) D hpD τ).choose + +lemma chosenFinset_card (D : ℕ) (hpD : D + 2 ≤ p) (τ : ZMod p) : + (chosenFinset (p := p) D hpD τ).card = D + 1 := + (exists_finset_card_avoiding (p := p) D hpD τ).choose_spec.1 + +lemma chosenFinset_not_mem (D : ℕ) (hpD : D + 2 ≤ p) (τ : ZMod p) : + τ ∉ chosenFinset (p := p) D hpD τ := + (exists_finset_card_avoiding (p := p) D hpD τ).choose_spec.2 + +variable {G₁ : Type} [Group G₁] [PrimeOrderWith G₁ p] {g₁ : G₁} + {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] {g₂ : G₂} + [∀ i, SampleableType (unifSpec.Range i)] + +/-- The winning ARSDH adversary. As with `t`-SDH it recovers `τ` from `g₂ ^ τ` in the verifier +leg of the SRS by `Classical.choice`, then returns the ARSDH solution: a size-`D+1` set `S` +avoiding `τ`, the nontrivial element `g₁`, and `g₁ ^ (1 / Z_S(τ))`. ZERO oracle queries. -/ +noncomputable def arsdhExtractingAdversary (hg₂ : g₂ ≠ 1) (D : ℕ) (hpD : D + 2 ≤ p) : + Groups.arsdhAdversary (G₁ := G₁) (G₂ := G₂) (p := p) D := + fun srs => + pure (some + (chosenFinset (p := p) D hpD (dlogOf (p := p) hg₂ srs.2[1]), + g₁, + g₁ ^ (1 / (∏ s ∈ chosenFinset (p := p) D hpD (dlogOf (p := p) hg₂ srs.2[1]), + (X - C s : CPolynomial (ZMod p))).eval (dlogOf (p := p) hg₂ srs.2[1])).val)) + +omit [PrimeOrderWith G₁ p] in +/-- The ARSDH game with the exhibited adversary collapses to a single `map` over the trapdoor +sampler: the adversary has already recovered `τ`, so `S`, `h₁`, `h₂` are functions of `τ`. -/ +lemma arsdh_game_run_eq (hg₂ : g₂ ≠ 1) (D : ℕ) (hpD : D + 2 ≤ p) : + (Groups.arsdhGame (g₁ := g₁) (g₂ := g₂) D + (arsdhExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D hpD)).run + = (fun τ : ZMod p => some + (τ, chosenFinset (p := p) D hpD τ, g₁, + g₁ ^ (1 / (∏ s ∈ chosenFinset (p := p) D hpD τ, + (X - C s : CPolynomial (ZMod p))).eval τ).val)) + <$> Groups.sampleNonzeroZMod := by + simp [Groups.arsdhGame, arsdhExtractingAdversary, Groups.PowerSrs.generate, + Groups.PowerSrs.tower, dlogOf_pow hg₂] + +/-- The exhibited adversary wins the ARSDH game with probability exactly `1`. -/ +theorem arsdhExperiment_arsdhExtractingAdversary (hg₁ : g₁ ≠ 1) (hg₂ : g₂ ≠ 1) + (D : ℕ) (hpD : D + 2 ≤ p) : + Groups.arsdhExperiment (g₁ := g₁) (g₂ := g₂) D + (arsdhExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D hpD) = 1 := by + classical + rw [Groups.arsdhExperiment, probEvent_eq_one_iff] + refine ⟨?_, ?_⟩ + · rw [OptionT.probFailure_eq, arsdh_game_run_eq (g₁ := g₁) hg₂ D hpD, probFailure_map, + probFailure_sampleNonzeroZMod] + simp + · intro x hx + rw [OptionT.support_def, arsdh_game_run_eq (g₁ := g₁) hg₂ D hpD, support_map] at hx + obtain ⟨τ, hτ, hxτ⟩ := hx + simp only [Option.some.injEq] at hxτ + subst hxτ + refine ⟨chosenFinset_card (p := p) D hpD τ, ?_, hg₁, rfl⟩ + exact KZG.CommitmentScheme.prod_x_sub_c_eval_ne_zero (chosenFinset_not_mem (p := p) D hpD τ) + +/-- **The refutation, for ARSDH.** ArkLib's `arsdhAssumption` is FALSE for every error bound +`< 1` (at every degree `D` with `p ≥ D + 2`, in every prime-order group pair with nontrivial +`g₁, g₂`). Same `Classical.choice` adversary, same argument as `not_tSdhAssumption`. -/ +theorem not_arsdhAssumption (hg₁ : g₁ ≠ 1) (hg₂ : g₂ ≠ 1) (D : ℕ) (hpD : D + 2 ≤ p) + (error : ℝ≥0) (herr : (error : ℝ≥0∞) < 1) : + ¬ Groups.arsdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) D error := by + intro h + have hle := h (arsdhExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D hpD) + rw [arsdhExperiment_arsdhExtractingAdversary (g₁ := g₁) hg₁ hg₂ D hpD] at hle + exact absurd (lt_of_le_of_lt hle herr) (lt_irrefl 1) + +omit [PrimeOrderWith G₂ p] in +/-- **The other error regime, for ARSDH.** For any error bound `≥ 1`, `arsdhAssumption` holds +trivially. Combined with `not_arsdhAssumption`, this exhausts the error regimes when +`D + 2 ≤ p`, which is the regime consumed by `function_binding`. -/ +theorem arsdhAssumption_trivial_of_one_le (D : ℕ) (error : ℝ≥0) + (herr : (1 : ℝ≥0∞) ≤ (error : ℝ≥0∞)) : + Groups.arsdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) D error := by + intro adversary + refine le_trans ?_ herr + rw [Groups.arsdhExperiment] + exact probEvent_le_one + +/-- CANARY: an ARSDH adversary that gives up loses with probability `1`, so `arsdhExperiment` +discriminates — the probability-`1` result above is about the exhibited adversary, not an +artifact of the machinery. -/ +def arsdhGivingUpAdversary (D : ℕ) : Groups.arsdhAdversary (G₁ := G₁) (G₂ := G₂) (p := p) D := + fun _ => pure none + +omit [PrimeOrderWith G₂ p] in +theorem arsdhExperiment_givingUpAdversary (D : ℕ) : + Groups.arsdhExperiment (g₁ := g₁) (g₂ := g₂) D + (arsdhGivingUpAdversary (G₁ := G₁) (G₂ := G₂) (p := p) D) = 0 := by + classical + rw [Groups.arsdhExperiment, probEvent_eq_zero_iff] + intro x hx + rw [OptionT.support_def] at hx + simp [Groups.arsdhGame, arsdhGivingUpAdversary] at hx + +/-- **Consumer.** `KZG.function_binding` derives evaluation/function binding from +`arsdhAssumption` under `hp : p ≥ n + 2` and `hpair : pairing g₁ g₂ ≠ 0`. Since that pairing +nondegeneracy forces `g₁ ≠ 1` and `g₂ ≠ 1` (see `g₂_ne_one_of_pairing_ne_zero`), and `p ≥ n + 2` +is exactly the hypothesis `not_arsdhAssumption` needs, `function_binding` is applicable only with +`arsdhError ≥ 1`, where its conclusion is the triviality `probability ≤ 1`. Identical vacuity to +`binding` / `t`-SDH. -/ +theorem arsdh_binding_hypotheses_unsatisfiable (hg₁ : g₁ ≠ 1) (hg₂ : g₂ ≠ 1) + (n : ℕ) (hp : n + 2 ≤ p) (arsdhError : ℝ≥0) (herr : (arsdhError : ℝ≥0∞) < 1) : + ¬ Groups.arsdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) n arsdhError := + not_arsdhAssumption (g₁ := g₁) hg₁ hg₂ n hp arsdhError herr + +end ArsdhRefutation + +end ArkLibVacuity + +#print axioms ArkLibVacuity.not_tSdhAssumption +#print axioms ArkLibVacuity.tSdhAssumption_trivial_of_one_le +#print axioms ArkLibVacuity.not_arsdhAssumption +#print axioms ArkLibVacuity.arsdhAssumption_trivial_of_one_le diff --git a/ArkLib/Scratch/KzgVacuity/README.md b/ArkLib/Scratch/KzgVacuity/README.md new file mode 100644 index 0000000000..b2d5d3f226 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/README.md @@ -0,0 +1,312 @@ +# KZG evaluation binding: a vacuity finding, an honest de-vacuation, and two GGM soundness bounds + +This directory (`ArkLib/Scratch/KzgVacuity/`) contains a mechanized formalization-soundness +finding about `ArkLib/Commitments/Functional/KZG/Binding.lean`, a minimal fix applied to that +file, and the generic-group soundness bound the fix points at — proved in **two** standard +generic-group models, **both wired to ArkLib's real `tSdhExperiment`**. Everything is checked +against ArkLib at `d72f8392ff03047dc5386f4f4bb513743e7ada65` (Lean `v4.31.0`, VCVio/CompPoly +`v4.31.0`), imports the genuine upstream modules, redefines nothing, builds `sorry`-free, and has +axiom closure exactly `[propext, Classical.choice, Quot.sound]`. + +**This is not a security advisory.** There is no vulnerability, nothing exploitable, and no +embargo. KZG, the `t`-SDH assumption as normally stated, and the reduction in `Binding.lean` +are all sound. The issue is the *quantifier* in one Lean assumption. We found the identical +pattern in our own Lean floors first; we bring it here as a shared field lesson, not a dunk. + +--- + +## 1. The finding: `tSdhAssumption` is vacuous, and so is `binding` + +`Groups.tSdhAssumption` quantifies over an *unrestricted* adversary type: + +```lean +def tSdhAssumption … (D : ℕ) (error : ℝ≥0) : Prop := + ∀ (adversary : tSdhAdversary D …), + tSdhExperiment (g₁ := g₁) (g₂ := g₂) D adversary ≤ (error : ℝ≥0∞) +``` + +`tSdhAdversary` lands in `StateT unifSpec.QueryCache ProbComp`. Because `ProbComp` is a free +monad over oracle queries, **pure computation is free** and no resource bound is imposed. An +adversary may therefore `pure` an arbitrary noncomputable function of the SRS at zero cost. + +The SRS includes the verifier leg `(g₂, g₂^τ)`, which determines `τ` whenever `g₂ ≠ 1`, and +ArkLib's own `Algebra.lean:105 exists_zmod_power_of_generator` makes that discrete log +`Classical.choice`-definable. So a one-line adversary recovers `τ`, returns the `t`-SDH +solution `(c = 0, g₁^{1/τ})`, and wins with probability *exactly* `1` (it makes zero oracle +queries). Consequently: + +- `tSdhAssumption D error` is **false for every `error < 1`** (`not_tSdhAssumption`), and +- trivially true for `error ≥ 1`, since a probability is `≤ 1` (`tSdhAssumption_trivial_of_one_le`). + +`KZG.CommitmentScheme.binding` takes `tSdhAssumption` as a hypothesis and concludes a bound +at the *same* `error`, so it carries no information at any parameter: below `1` its premise is +unsatisfiable, at or above `1` its conclusion is free. `binding`'s own `hpair : pairing g₁ g₂ ≠ 0` +even *forces* `g₂ ≠ 1` (via bilinearity and `map_zero`), so the killing adversary's one +hypothesis is discharged from `binding`'s own premises (`binding_hypotheses_unsatisfiable`). + +The sibling `Groups.arsdhAssumption` — the hypothesis of `KZG.function_binding` — has the +identical unrestricted quantifier and falls the identical way (`not_arsdhAssumption` / +`arsdhAssumption_trivial_of_one_le`); the ARSDH branch's `D + 2 ≤ p` is exactly the +`p ≥ n + 2` that `function_binding` already carries. + +The mechanized witness is `KzgVacuity.lean` (namespace `ArkLibVacuity`). It ships with +canaries — `tSdhExperiment_givingUpAdversary = 0`, `arsdhExperiment_givingUpAdversary = 0` — +proving the `= 1` is a fact about *this* adversary, not an artifact of the probability +machinery: the experiment genuinely discriminates. + +The idiom is not special to `t`-SDH. Stated in ArkLib's adversary type, a `q`-strong-DLOG base +assumption ("recover the SRS trapdoor `τ` from the KZG power-SRS") is vacuous the same way, by +the identical `Classical.choice` extraction — so "reduce KZG binding to `q`-DLOG instead" does +not, by itself, escape the hole: any base assumption must first be stated over a sound +adversary class. + +**Why `#print axioms` does not catch this.** `binding` is axiom-clean *and* vacuous at the +same time. A clean axiom closure certifies "no `sorry`, no `native_decide`"; it says nothing +about whether a hypothesis is satisfiable. That blindness is the whole reason the pattern is +easy to miss, and it is why we treat it as a discipline problem rather than a typo. + +--- + +## 2. The fix: an extraction-shaped restatement (`+42 / −14`, one file) + +The right tool here is **not** query-bounding. `t`-SDH is an *algebraic* assumption whose +killing adversary makes zero queries, so an `IsQueryBoundP`-style restriction (the correct +fix for random-oracle/hash floors) constrains something this adversary never does. The honest +menu is the generic/algebraic group model, or an extraction-shaped restatement that turns the +assumption into *data the adversary must produce*. We ship the latter as the minimal step — +it is the pattern VCVio already uses for its Merkle `Binding`. + +The key observation is structural: **ArkLib's reduction is already fully constructive.** +`binding`'s proof is a five-step `calc`; the first four steps are unconditional transition +lemmas, and `tSdhAssumption` is consumed in exactly one place — the last `≤`. So the fix is to +*split the calc at that last step*: + +```lean +/-- Extraction-shaped evaluation binding: every binding adversary yields — as the explicit + reduction `bindingReduction … adversary` — a t-SDH adversary whose success probability + upper-bounds its binding advantage. No assumption `Prop`, hence nothing for a + `Classical.choice` adversary to inhabit. -/ +theorem binding_reduces_to_tSdh {g₁ : G₁} {g₂ : G₂} (hg₁ : g₁ ≠ 1) + (hpair : pairing g₁ g₂ ≠ 0) [SampleableType G₁] (AuxState : Type) + (adversary : KzgBindingAdversary p G₁ G₂ n unifSpec AuxState) : + Commitment.bindingExperiment … (kzg …) AuxState adversary + ≤ Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) n + (bindingReduction … AuxState adversary) := by + … -- the existing calc prefix, verbatim; the four transition lemmas are untouched + +/-- The original assumption-form binding, now a one-line corollary. -/ +theorem binding … (htSdh : Groups.tSdhAssumption … n tSdhError) : + Commitment.binding … (kzg …) tSdhError := by + simp only [Commitment.binding]; intro AuxState adversary + exact (binding_reduces_to_tSdh (pairing := pairing) hg₁ hpair AuxState adversary).trans + (t_sdh_error_bound … tSdhError htSdh adversary) +``` + +`binding_reduces_to_tSdh` carries the full constructive content *without* the +universally-quantified assumption, so it is immune to the vacuity: the bound relates two +concrete probabilities — *this* adversary's advantage and *its* reduction's success — and +carries content at every parameter. `binding` keeps its exact signature (backward +compatible) as an immediate corollary; its docstring notes that the corollary only becomes +informative once `tSdhAssumption` is stated over a restricted adversary class. + +The full diff is `+42 / −14` in `Binding.lean`. The whole tree still builds; both theorems are +`[propext, Classical.choice, Quot.sound]`. + +### The fix survives the exact attack (mechanized) + +A de-vacuation is only honest if it survives the attack rather than merely avoiding it. +`RepairSurvives.lean` (namespace `ArkLibRepairCheck`) proves both facts as one conjunction, +`repair_survives_attack`: + +1. the identical trapdoor-extracting adversary *still* refutes `tSdhAssumption` below `1` + (we did not weaken the assumption), **and** +2. `binding_reduces_to_tSdh` holds *unconditionally* — it takes no `tSdhAssumption` + hypothesis, so leg (1) has nothing to empty. + +Both hold at once, in the same groups, `sorry`-free. That is the precise sense in which the +vacuity is closed: the disease was an unsatisfiable premise; the cure removes the premise +while keeping every step of the reduction. + +--- + +## 3. The sound numeric bound: KZG binding in the generic group model, two ways — both wired + +The extraction-shaped fix removes the vacuity but hands *no number*: its right-hand side is +still `tSdhExperiment` of the constructed reduction adversary. The number a KZG binding bound +ultimately rests on is the generic-group hardness of `t`-SDH. This directory mechanizes it in +the **two** standard generic-group models — Maurer's explicit-equality model [Mau05] and +Shoup's random-encoding model [Sho97] — and **wires both to ArkLib's real `tSdhExperiment`**. +Both yield the same Boneh–Boyen [BB04] numerator `C(fuel+D+4, 2)·D + (D+1)` over `p − 1`. + +Group elements are modelled as opaque handles carrying *ordinary* polynomials in the trapdoor +indeterminate `X` (**not Laurent** — group inversion negates the exponent, it does not +introduce `X⁻¹`; that is exactly why a winning `1/(X+c)` output is unrepresentable and forces +a bounded-degree root event). The oracle is pairing-free (`lin` moves only), matching ArkLib's +`G₁`-only `tSdhAdversary`, so the honest collision degree is `δ = D`. + +### 3a. Maurer explicit-equality track — wired via `embed` + +The Maurer model spends a `Move.query` step per equality test; only queried pairs enter the +bad event. The capstone is `GgmEndToEnd.tSdh_ggm_sound`, stated about ArkLib's **real** +`tSdhExperiment` via the embedding `embed`: + +```lean +theorem tSdh_ggm_sound … (strat : Strat p) (fuel : ℕ) : + tSdhExperiment D (embed strat) ≤ (C(fuel+D+4,2)·D + (D+1)) / (p − 1) +``` + +with a companion `tSdh_ggm_sound_lt_one` giving a genuine `< 1` in the standard regime +`C(fuel+D+4,2)·D + (D+1) < p − 1` (at cryptographic parameters, `≈ 2⁻²³⁴`). It quantifies over +the **image of the generic embedding** `embed` — the generic-restricted class that escapes the +vacuity: `embed strat` receives only equality booleans, never a group element, so it can only +realize `g₁^{f(τ)}` with `deg f ≤ D`, which is exactly what the counting bound bounds. The full +`tSdhAdversary` type does *not* escape (§1 proves the statement over it false); the embedding is +what makes the number meaningful. + +### 3b. Shoup random-encoding track — also wired via `embedShoup` + +The Shoup model gives the adversary **free** comparison: at every step it observes the full +pairwise-equality matrix of all its held encodings and branches on the entire pattern-history, +at no fuel or handle cost. This is the genuinely different model — the all-pairs collision +event is now *tight* rather than a conservative over-count. It is proved in two layers: + +* `GgmShoup.shoup_ggm_sound` states the bound about the free-comparison experiment + `shoupExperiment` on that model's own SRS seeding — the model-internal statement, where the + crux (an identical-until-bad hybrid over a whole equality matrix, `runShoup_congr_off_bad`, + discharged from a single global non-collision fact) is proved rather than assumed. +* `GgmShoupEmbed.shoup_tSdh_ggm_sound` **wires that model into ArkLib's real `tSdhExperiment`**, + exactly as the Maurer track is wired via `embed`: + +```lean +theorem shoup_tSdh_ggm_sound (hord₁ : orderOf g₁ = p) (hD : 1 ≤ D) + (strat : ShoupStrat p) (fuel : ℕ) : + tSdhExperiment (g₁ := g₁) (g₂ := g₂) D (embedShoup g₁ D fuel strat) + ≤ (C(fuel+D+4,2)·D + (D+1)) / (p − 1) +``` + +with a companion `shoup_tSdh_ggm_sound_lt_one`. Free comparison is **realized, not assumed**: +a real `tSdhAdversary` holds actual `G₁` elements and can test equality of any two it holds for +free (`DecidableEq G₁`, classically). In a prime-order group the exponent encoding +`a ↦ g₁^{a.val}` is injective (`GgmArkLibTransport.gpow_val_inj_iff`), so the full pairwise +group-equality matrix of the adversary's realized handles equals the symbolic +`eqPattern (realAns τ)` the strategy branches on — discharged off the bad event by +`groupEqPattern_eq`. The lazily-sampled encoding `σ : ZMod p ↪ E` never enters the +mechanization: injectivity folds it away, exactly as `gpow_val_inj_iff` folds the concrete +encoding away in the Maurer embed. + +The numerator is byte-identical to the Maurer track's; the difference is the *model in which it +is proved*. With both `embed` and `embedShoup` in place, the two standard GGM formulations bound +the **same** real experiment through one socket each. + +### The dependency spine + +All `sorry`-free, axioms exactly `[propext, Classical.choice, Quot.sound]`: + +| Module | Role | +|---|---| +| `GgmCandidate` | static (zero-query) Schwartz–Zippel core, `(D+1)/(p−1)`; reused by both tracks | +| `GgmDegreeInvariant` | structural handle-table degree invariants (`natDegree_getD_le`, …); reused by both tracks | +| `GgmAdaptive` | the adaptive `q`-query bound; identical-until-bad hybrid by induction on fuel | +| `GgmRandomEncoding` | the all-pairs (quadratic) collision count at `δ = D`; table size is a theorem | +| `GgmArkLibTransport` | field→group transport against ArkLib's real `Groups.tSdhCondition` | +| `GgmProbThreading` | collapses ArkLib's `OptionT ProbComp` / `StateT QueryCache` game to `card/(p−1)` | +| `GgmDegreeDischarge` | discharges the SRS degree invariant on the *actual* (linear, pairing-free) oracle | +| `GgmEmbed` | constructs the generic-restricted Maurer adversary and certifies what it realizes | +| `GgmEndToEnd` | the Maurer capstone `tSdh_ggm_sound` (+ `tSdh_ggm_sound_lt_one`), wired to `tSdhExperiment` | +| `GgmShoup` | the Shoup free-comparison capstone `shoup_ggm_sound` (+ `_lt_one`), model-internal | +| `GgmShoupEmbed` | wires Shoup into `tSdhExperiment`: `shoup_tSdh_ggm_sound` (+ `_lt_one`) | + +To our knowledge — a census of ArkLib, VCVio, and Mathlib — no generic-group-model security +*theorem* previously existed in Lean, so this is a candidate first of its kind. ArkLib's own +`AGM/Basic.lean` is a WIP stub (`Adversary.run` is `sorry`, zero theorems, orphaned) and is +moreover unsound as written: its adversary is a `ReaderT` over the concrete group table, so its +outputs can still depend on discrete logs. If you would prefer to complete that module to +opacity instead, the extraction-shaped fix in §2 is the right first step regardless: it isolates +the *single* obligation (bound the success of the one reduction adversary) that any restricted +assumption — generic, algebraic, or otherwise — must discharge. + +### Honest side-conditions on both bounds + +These travel with every citation: + +- `1 ≤ D` — the meaningful KZG regime; at `D = 0` a pairing-free `G₁` adversary genuinely + cannot form `g₁^τ`. +- `2 ≤ p` (so `p − 1 ≥ 1`) and, for both tracks' transport into `tSdhExperiment`, + `orderOf g₁ = p` (the base is a generator, used for encoding injectivity). +- Maurer only: `[∀ i, SampleableType (unifSpec.Range i)]` — ArkLib's own instance on + `tSdhExperiment`, carried verbatim. +- The bound is the classical Boneh–Boyen shape `O((q_G + D)²·D / p)` — degree-dependent, **not** + a clean `q²/p`. +- Both capstones quantify over the **image of their embedding** (`embed` / `embedShoup`) into + ArkLib's `tSdhAdversary`, not the full type — over the full type the statement is false (§1). + `GgmShoup.shoup_ggm_sound` is the model-internal statement about `shoupExperiment`; + `GgmShoupEmbed.shoup_tSdh_ggm_sound` is the one wired to ArkLib's `tSdhExperiment`. + +`GgmRandomEncoding` additionally carries, clearly labelled as **off-path**, a conservative +pairing-capable `δ = 2D` variant (`rand_encoding_bound` / `_srs` / `card_pairRootUnion_le_two_mul`). +It is not consumed by either capstone — both take the `δ = D` chain — and is kept only as the +conservative ceiling for a stronger, off-interface (pairing-capable) adversary. It builds +`sorry`-free on the same axioms. + +--- + +## 4. Build and check + +Against ArkLib at `d72f8392` with Lean `v4.31.0` (VCVio/CompPoly `v4.31.0`), the modules under +`ArkLib/Scratch/KzgVacuity/` (these are `Scratch` modules — build them by explicit target; the +default `lake build` builds the main `ArkLib` library, which includes the §2 fix to +`Binding.lean`): + +```bash +# The finding +lake build ArkLib.Scratch.KzgVacuity.KzgVacuity +# #print axioms ArkLibVacuity.not_tSdhAssumption → [propext, Classical.choice, Quot.sound] +# #print axioms ArkLibVacuity.not_arsdhAssumption → [propext, Classical.choice, Quot.sound] +# #print axioms ArkLibVacuity.tSdhAssumption_trivial_of_one_le → [propext, Classical.choice, Quot.sound] + +# The fix (applied to ArkLib/Commitments/Functional/KZG/Binding.lean) and its survival proof +lake build ArkLib.Scratch.KzgVacuity.RepairSurvives +# #print axioms KZG.CommitmentScheme.binding_reduces_to_tSdh → [propext, Classical.choice, Quot.sound] +# #print axioms ArkLibRepairCheck.repair_survives_attack → [propext, Classical.choice, Quot.sound] + +# The GGM bound, Maurer track (wired to ArkLib's real tSdhExperiment) +lake build ArkLib.Scratch.KzgVacuity.GgmEndToEnd +# #print axioms GgmEndToEnd.tSdh_ggm_sound → [propext, Classical.choice, Quot.sound] +# #print axioms GgmEndToEnd.tSdh_ggm_sound_lt_one → [propext, Classical.choice, Quot.sound] + +# The GGM bound, Shoup track (wired to ArkLib's real tSdhExperiment) +lake build ArkLib.Scratch.KzgVacuity.GgmShoupEmbed +# #print axioms GgmShoup.shoup_ggm_sound → [propext, Classical.choice, Quot.sound] +# #print axioms GgmShoupEmbed.shoup_tSdh_ggm_sound → [propext, Classical.choice, Quot.sound] +# #print axioms GgmShoupEmbed.shoup_tSdh_ggm_sound_lt_one → [propext, Classical.choice, Quot.sound] +``` + +Every headline theorem is `sorry`-free with axiom closure exactly +`[propext, Classical.choice, Quot.sound]` — no `sorryAx`, no `native_decide`, no +`ofReduceBool`. The `Classical.choice` in the vacuity theorems is the *content*, not a smell: +it is the unbounded extractor being exhibited as a legal inhabitant of the unrestricted +adversary type. + +--- + +## 5. A note on framing + +We ran this same "try to prove each hardness floor false at its deployed parameters" tooth on +our own Lean tree before pointing it here, and found the identical unrestricted-quantifier hole +in several of our own floors first. `#print axioms` was blind to all of them. The reduction in +`Binding.lean` is careful, correct work — which is exactly why the honest thing is to state it +soundly rather than route around it. We bring the finding, the fix, and both GGM bounds together +so the whole story is one branch a reviewer can `git checkout` at any commit and `lake build`. + +## References + +- **[BB04]** Boneh, D., and Boyen, X. *Short Signatures Without Random Oracles.* EUROCRYPT 2004. +- **[KZG10]** Kate, A., Zaverucha, G. M., and Goldberg, I. *Constant-Size Commitments to + Polynomials and Their Applications.* ASIACRYPT 2010. +- **[Sho97]** Shoup, V. *Lower Bounds for Discrete Logarithms and Related Problems.* EUROCRYPT 1997. +- **[Mau05]** Maurer, U. *Abstract Models of Computation in Cryptography.* IMA 2005. +- **[FKL18]** Fuchsbauer, G., Kiltz, E., and Loss, J. *The Algebraic Group Model and its + Applications.* CRYPTO 2018. +- **[Sch80]** Schwartz, J. T. *Fast Probabilistic Algorithms for Verification of Polynomial + Identities.* J. ACM 1980. +- **[Zip79]** Zippel, R. *Probabilistic Algorithms for Sparse Polynomials.* EUROSAM 1979. diff --git a/ArkLib/Scratch/KzgVacuity/RepairSurvives.lean b/ArkLib/Scratch/KzgVacuity/RepairSurvives.lean new file mode 100644 index 0000000000..be58b2d579 --- /dev/null +++ b/ArkLib/Scratch/KzgVacuity/RepairSurvives.lean @@ -0,0 +1,206 @@ +/- +Copyright (c) 2026 Ember Arlynx. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Ember Arlynx +-/ +import ArkLib.Commitments.Functional.KZG.Binding + +/-! +# The extraction-shaped binding repair survives the vacuity attack + +The companion `KzgVacuity` file shows that ArkLib's `Groups.tSdhAssumption` [BB04], as stated over +concrete groups, is refuted by a `Classical.choice`-definable trapdoor extractor. This file shows +that the *extraction-shaped* repair of KZG [KZG10] evaluation binding — +`KZG.CommitmentScheme.binding_reduces_to_tSdh` — survives that exact attack. + +The refutation section (`dlogOf` … `not_tSdhAssumption`) is re-proved here against the repaired +`Binding.lean`, so the attack and the repair coexist in one axiom closure. The final theorem +`repair_survives_attack` states, for the same groups and pairing in which the attack succeeds: + +1. the trapdoor-extracting adversary still refutes `tSdhAssumption` below error $1$ + (`not_tSdhAssumption`); and +2. the repaired reduction bound `binding_reduces_to_tSdh` holds unconditionally and relates two + concrete probabilities — it never mentions `tSdhAssumption`, so there is nothing for the + choice-adversary to inhabit. + +The repair reduces binding to a bound between probabilities rather than to an assumption whose +statement quantifies over all concrete-group adversaries, which is where the vacuity lived. + +## References + +* [Boneh, D., and Boyen, X., *Short Signatures Without Random Oracles*][BB04] +* [Kate, A., Zaverucha, G. M., and Goldberg, I., *Constant-Size Commitments to Polynomials and + Their Applications*][KZG10] +-/ + +open OracleSpec OracleComp +open scoped NNReal ENNReal + +namespace ArkLibRepairCheck + +section Dlog + +variable {p : ℕ} [Fact (Nat.Prime p)] + +/-- The choice-definable discrete logarithm base a nontrivial `g` in a prime-order group: +`Exists.choose` applied to ArkLib's own `Groups.exists_zmod_power_of_generator`. -/ +noncomputable def dlogOf {G : Type} [Group G] [PrimeOrderWith G p] {g : G} (hg : g ≠ 1) + (x : G) : ZMod p := + (Groups.exists_zmod_power_of_generator (G := G) PrimeOrderWith.hCard hg + (Groups.orderOf_eq_prime_of_ne_one g hg) x).choose + +lemma dlogOf_pow {G : Type} [Group G] [PrimeOrderWith G p] {g : G} (hg : g ≠ 1) (a : ZMod p) : + dlogOf (p := p) hg (g ^ a.val) = a := by + have hord : orderOf g = p := Groups.orderOf_eq_prime_of_ne_one g hg + have hspec : g ^ a.val = g ^ (dlogOf (p := p) hg (g ^ a.val)).val := + (Groups.exists_zmod_power_of_generator (G := G) PrimeOrderWith.hCard hg hord + (g ^ a.val)).choose_spec + have hdiv : g ^ (dlogOf (p := p) hg (g ^ a.val) - a).val = 1 := by + rw [← Groups.gpow_div_eq hord _ a, ← hspec, div_self'] + exact sub_eq_zero.mp (Groups.zmod_eq_zero_of_gpow_eq_one hord hdiv) + +lemma sampleNonzeroZMod_ne_zero {τ : ZMod p} + (hτ : τ ∈ support (Groups.sampleNonzeroZMod (p := p))) : τ ≠ 0 := by + have hp : 1 < p := Nat.Prime.one_lt Fact.out + haveI : NeZero (p - 1) := ⟨Nat.pos_iff_ne_zero.mp (Nat.sub_pos_of_lt hp)⟩ + haveI : NeZero p := ⟨Nat.pos_iff_ne_zero.mp (Nat.zero_lt_of_lt hp)⟩ + rw [Groups.sampleNonzeroZMod, support_map] at hτ + obtain ⟨i, -, rfl⟩ := hτ + have hi := i.isLt + have hlt : (i : ℕ) + 1 < p := by omega + intro hzero + simp only at hzero + have hdvd : (((i : ℕ) + 1 : ℕ) : ZMod p) = 0 := by push_cast; exact hzero + rw [ZMod.natCast_eq_zero_iff] at hdvd + exact absurd (Nat.le_of_dvd (Nat.succ_pos _) hdvd) (not_le.mpr hlt) + +lemma probFailure_sampleNonzeroZMod : Pr[⊥ | Groups.sampleNonzeroZMod (p := p)] = 0 := by + rw [Groups.sampleNonzeroZMod]; simp + +end Dlog + +section Refutation + +-- `PrimeOrderWith G₁ p` is deliberately absent: the t-SDH solution lives in `G₁` as a bare +-- group element, so nothing in this section needs `G₁` prime-order. +variable {p : ℕ} [Fact (Nat.Prime p)] + {G₁ : Type} [Group G₁] {g₁ : G₁} + {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] {g₂ : G₂} + [∀ i, SampleableType (unifSpec.Range i)] + +/-- The winning t-SDH adversary: reads `g₂ ^ τ` from the verifier SRS leg, recovers `τ` by +`Classical.choice`, returns the solution at offset `c = 0`. Zero oracle queries. -/ +noncomputable def tauExtractingAdversary (hg₂ : g₂ ≠ 1) (D : ℕ) : + Groups.tSdhAdversary (G₁ := G₁) (G₂ := G₂) (p := p) D := + fun srs => pure (some (0, g₁ ^ (1 / dlogOf (p := p) hg₂ srs.2[1]).val)) + +lemma game_run_eq (hg₂ : g₂ ≠ 1) (D : ℕ) : + (Groups.tSdhGame (g₁ := g₁) (g₂ := g₂) D + (tauExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D)).run + = (fun τ : ZMod p => some (τ, (0 : ZMod p), g₁ ^ (1 / τ).val)) + <$> Groups.sampleNonzeroZMod := by + simp [Groups.tSdhGame, tauExtractingAdversary, Groups.PowerSrs.generate, + Groups.PowerSrs.tower, dlogOf_pow hg₂] + +theorem tSdhExperiment_tauExtractingAdversary (hg₂ : g₂ ≠ 1) (D : ℕ) : + Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) D + (tauExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D) = 1 := by + classical + rw [Groups.tSdhExperiment, probEvent_eq_one_iff] + refine ⟨?_, ?_⟩ + · rw [OptionT.probFailure_eq, game_run_eq (g₁ := g₁) hg₂ D, probFailure_map, + probFailure_sampleNonzeroZMod] + simp + · intro x hx + rw [OptionT.support_def, game_run_eq (g₁ := g₁) hg₂ D, support_map] at hx + obtain ⟨τ, hτ, hxτ⟩ := hx + simp only [Option.some.injEq] at hxτ + subst hxτ + have hτ0 : τ ≠ 0 := sampleNonzeroZMod_ne_zero hτ + exact ⟨by simpa using hτ0, by simp⟩ + +/-- **The exact attack.** ArkLib's `tSdhAssumption` is FALSE for every error bound `< 1`. -/ +theorem not_tSdhAssumption (hg₂ : g₂ ≠ 1) (D : ℕ) (error : ℝ≥0) (herr : (error : ℝ≥0∞) < 1) : + ¬ Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) D error := by + intro h + have hle := h (tauExtractingAdversary (G₁ := G₁) (g₁ := g₁) (g₂ := g₂) (p := p) hg₂ D) + rw [tSdhExperiment_tauExtractingAdversary (g₁ := g₁) hg₂ D] at hle + exact absurd (lt_of_le_of_lt hle herr) (lt_irrefl 1) + +end Refutation + +section RepairSurvives + +variable {p : ℕ} [Fact (Nat.Prime p)] + {G₁ : Type} [Group G₁] [PrimeOrderWith G₁ p] {g₁ : G₁} + {G₂ : Type} [Group G₂] [PrimeOrderWith G₂ p] {g₂ : G₂} + {Gₜ : Type} [Group Gₜ] [PrimeOrderWith Gₜ p] [DecidableEq Gₜ] + [Module (ZMod p) (Additive G₁)] [Module (ZMod p) (Additive G₂)] + [Module (ZMod p) (Additive Gₜ)] + +variable {n : ℕ} + +open CompPoly CompPoly.CPolynomial in +/-- Mirror of ArkLib's `local instance bindingOracleInterface` (Binding.lean:51). Because +that instance is `local`, it is not in scope here, and the KZG binding types would otherwise +resolve `OracleInterface (Fin (n+1) → ZMod p)` to the generic `instFunction`, mismatching the +instance the library was compiled against. Declaring the identical local instance makes the +types line up. -/ +local instance bindingOracleInterface : OracleInterface (Fin (n + 1) → ZMod p) where + Query := ZMod p + toOC.spec := ZMod p →ₒ ZMod p + toOC.impl z := do return (CPolynomial.ofFn (← read)).eval z + +omit [DecidableEq Gₜ] in +/-- `binding`'s pairing hypothesis forces `g₂ ≠ 1` (bilinear pairing kills the identity). -/ +lemma g₂_ne_one_of_pairing_ne_zero + (pairing : (Additive G₁) →ₗ[ZMod p] (Additive G₂) →ₗ[ZMod p] (Additive Gₜ)) + (hpair : pairing (Additive.ofMul g₁) (Additive.ofMul g₂) ≠ 0) : g₂ ≠ 1 := by + intro h + apply hpair + rw [show (Additive.ofMul g₂) = 0 from congrArg Additive.ofMul h] + exact map_zero _ + +/-- **The repair survives the exact attack.** For any prime-order group pair and any +nondegenerate pairing — precisely the setting in which the trapdoor-extracting adversary +refutes `tSdhAssumption` — BOTH of the following hold simultaneously: + +* `(1)` the exact attack still refutes the assumption below error `1` + (`not_tSdhAssumption`); and +* `(2)` the repaired, extraction-shaped reduction bound + `KZG.CommitmentScheme.binding_reduces_to_tSdh` holds *unconditionally*, upper-bounding + every binding adversary's advantage by the success probability of its explicit t-SDH + reduction. + +The old assumption-form `binding` was vacuous because `(1)` made its premise false. The new +form is not: `(2)` never takes `tSdhAssumption` as a hypothesis, so `(1)` cannot empty it. +The two live together in one `sorry`-free axiom closure. -/ +theorem repair_survives_attack + (pairing : (Additive G₁) →ₗ[ZMod p] (Additive G₂) →ₗ[ZMod p] (Additive Gₜ)) + (hg₁ : g₁ ≠ 1) + (hpair : pairing (Additive.ofMul g₁) (Additive.ofMul g₂) ≠ 0) + [SampleableType G₁] + (tSdhError : ℝ≥0) (herr : (tSdhError : ℝ≥0∞) < 1) + (AuxState : Type) + (adversary : KZG.CommitmentScheme.KzgBindingAdversary p G₁ G₂ n unifSpec AuxState) : + (¬ Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) n tSdhError) + ∧ (Commitment.bindingExperiment (init := pure ∅) (impl := randomOracle) + (KZG.CommitmentScheme.kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing)) + AuxState adversary + ≤ Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) n + (KZG.CommitmentScheme.bindingReduction (g₁ := g₁) (g₂ := g₂) (pairing := pairing) + AuxState adversary)) := by + refine ⟨?_, ?_⟩ + · exact not_tSdhAssumption (g₁ := g₁) + (g₂_ne_one_of_pairing_ne_zero pairing hpair) n tSdhError herr + · exact KZG.CommitmentScheme.binding_reduces_to_tSdh (pairing := pairing) hg₁ hpair + AuxState adversary + +end RepairSurvives + +end ArkLibRepairCheck + +#print axioms ArkLibRepairCheck.not_tSdhAssumption +#print axioms ArkLibRepairCheck.repair_survives_attack +#print axioms KZG.CommitmentScheme.binding_reduces_to_tSdh +#print axioms KZG.CommitmentScheme.binding diff --git a/blueprint/src/references.bib b/blueprint/src/references.bib index f841ee5425..93fa484964 100644 --- a/blueprint/src/references.bib +++ b/blueprint/src/references.bib @@ -510,3 +510,74 @@ @inproceedings{NS24 year = {2024}, publisher = {Springer} } + +@inproceedings{BB04, + author = {Dan Boneh and Xavier Boyen}, + title = {Short Signatures Without Random Oracles}, + booktitle = {Advances in Cryptology -- EUROCRYPT 2004}, + series = {Lecture Notes in Computer Science}, + volume = {3027}, + pages = {56--73}, + publisher = {Springer}, + year = {2004}, + doi = {10.1007/978-3-540-24676-3_4} +} + +@inproceedings{Sho97, + author = {Victor Shoup}, + title = {Lower Bounds for Discrete Logarithms and Related Problems}, + booktitle = {Advances in Cryptology -- EUROCRYPT 1997}, + series = {Lecture Notes in Computer Science}, + volume = {1233}, + pages = {256--266}, + publisher = {Springer}, + year = {1997}, + doi = {10.1007/3-540-69053-0_18} +} + +@inproceedings{Mau05, + author = {Ueli Maurer}, + title = {Abstract Models of Computation in Cryptography}, + booktitle = {Cryptography and Coding, 10th IMA International Conference}, + series = {Lecture Notes in Computer Science}, + volume = {3796}, + pages = {1--12}, + publisher = {Springer}, + year = {2005}, + doi = {10.1007/11586821_1} +} + +@inproceedings{FKL18, + author = {Georg Fuchsbauer and Eike Kiltz and Julian Loss}, + title = {The Algebraic Group Model and its Applications}, + booktitle = {Advances in Cryptology -- CRYPTO 2018}, + series = {Lecture Notes in Computer Science}, + volume = {10992}, + pages = {33--62}, + publisher = {Springer}, + year = {2018}, + doi = {10.1007/978-3-319-96881-0_2} +} + +@article{Sch80, + author = {Jacob T. 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