perf: using non-native Eval for curve arithmetic

[yelhousni]

Description

Similar to #1312 but for sw_emulated and sw_bw6761 curve arithmetic.

Type of change

• New feature (non-breaking change which adds functionality)

How has this been tested?

Current tests pass.

How has this been benchmarked?

• Emulated PLONK recursion (BW6-761 in BN254):

old	new	diff
21546468	19241163	2305305

• ECRECOVER precompile:

old	new	diff
428288	366181	62107

• ECMUL precompile:

old	new	diff
261293	221844	39449

Checklist:

• I have performed a self-review of my code
• I have commented my code, particularly in hard-to-understand areas
• I have made corresponding changes to the documentation
• I have added tests that prove my fix is effective or that my feature works
• I did not modify files generated from templates
• golangci-lint does not output errors locally
• New and existing unit tests pass locally with my changes
• Any dependent changes have been merged and published in downstream modules

[ivokub]

Suggested edit:

diff --git a/std/algebra/emulated/sw_bw6761/g1.go b/std/algebra/emulated/sw_bw6761/g1.go
index afdf15d9..02278f42 100644
--- a/std/algebra/emulated/sw_bw6761/g1.go
+++ b/std/algebra/emulated/sw_bw6761/g1.go
@@ -75,7 +75,7 @@ func (g1 *G1) double(p *G1Affine) *G1Affine {
 	mone := g1.curveF.NewElement(-1)
 	xr := g1.curveF.Eval([][]*baseEl{{λ, λ}, {mone, &p.X}}, []int{1, 2})
 
-	// yr = λ(p-xr) - p.y
+	// yr = λ(p.x-xr) - p.y
 	yr := g1.curveF.Eval([][]*baseEl{{λ, &p.X}, {mone, λ, xr}, {mone, &p.Y}}, []int{1, 1, 1})
 
 	return &G1Affine{
diff --git a/std/algebra/emulated/sw_bw6761/g2.go b/std/algebra/emulated/sw_bw6761/g2.go
index e46eb120..0709eb5d 100644
--- a/std/algebra/emulated/sw_bw6761/g2.go
+++ b/std/algebra/emulated/sw_bw6761/g2.go
@@ -151,7 +151,7 @@ func (g2 *G2) double(p *G2Affine) *G2Affine {
 	mone := g2.curveF.NewElement(-1)
 	xr := g2.curveF.Eval([][]*baseEl{{λ, λ}, {mone, &p.P.X}}, []int{1, 2})
 
-	// yr = λ(p-xr) - p.y
+	// yr = λ(p.x-xr) - p.y
 	yr := g2.curveF.Eval([][]*baseEl{{λ, &p.P.X}, {mone, λ, xr}, {mone, &p.P.Y}}, []int{1, 1, 1})
 
 	return &G2Affine{

[ivokub]

Suggested edit:

diff --git a/std/algebra/emulated/sw_bw6761/pairing.go b/std/algebra/emulated/sw_bw6761/pairing.go
index 04eec856..03fab1c5 100644
--- a/std/algebra/emulated/sw_bw6761/pairing.go
+++ b/std/algebra/emulated/sw_bw6761/pairing.go
@@ -19,7 +19,6 @@ type Pairing struct {
 	curve  *sw_emulated.Curve[BaseField, ScalarField]
 	g1     *G1
 	g2     *G2
-	g2gen  *G2Affine
 }
 
 type GTEl = fields_bw6761.E6
@@ -63,15 +62,6 @@ func NewPairing(api frontend.API) (*Pairing, error) {
 	}, nil
 }
 
-func (pr Pairing) generators() *G2Affine {
-	if pr.g2gen == nil {
-		_, _, _, g2gen := bw6761.Generators()
-		cg2gen := NewG2AffineFixed(g2gen)
-		pr.g2gen = &cg2gen
-	}
-	return pr.g2gen
-}
-
 // FinalExponentiation computes the exponentiation zᵈ where
 //
 // d = (p⁶-1)/r = (p⁶-1)/Φ₆(p) ⋅ Φ₆(p)/r = (p³-1)(p+1)(p²-p+1)/r
@@ -436,53 +426,6 @@ func (pr Pairing) doubleAndAddStep(p1, p2 *g2AffP, isSub bool) (*g2AffP, *lineEv
 	return &p, &line1, &line2
 }
 
-// doubleAndSubStep doubles p1 and subs p2 to the result in affine coordinates, and evaluates the line in Miller loop
-// https://eprint.iacr.org/2022/1162 (Section 6.1)
-func (pr Pairing) doubleAndSubStep(p1, p2 *g2AffP) (*g2AffP, *lineEvaluation, *lineEvaluation) {
-
-	var line1, line2 lineEvaluation
-	var p g2AffP
-	mone := pr.curveF.NewElement(-1)
-
-	// compute λ1 = (y2-y1)/(x2-x1)
-	n := pr.curveF.Add(&p1.Y, &p2.Y)
-	d := pr.curveF.Sub(&p1.X, &p2.X)
-	l1 := pr.curveF.Div(n, d)
-
-	// compute x3 =λ1²-x1-x2
-	x3 := pr.curveF.Eval([][]*baseEl{{l1, l1}, {mone, &p1.X}, {mone, &p2.X}}, []int{1, 1, 1})
-
-	// omit y3 computation
-
-	// compute line1
-	line1.R0 = *l1
-	line1.R1 = *pr.curveF.Mul(l1, &p1.X)
-	line1.R1 = *pr.curveF.Sub(&line1.R1, &p1.Y)
-
-	// compute λ2 = -λ1-2y1/(x3-x1)
-	n = pr.curveF.MulConst(&p1.Y, big.NewInt(2))
-	d = pr.curveF.Sub(x3, &p1.X)
-	l2 := pr.curveF.Div(n, d)
-	l2 = pr.curveF.Add(l2, l1)
-	l2 = pr.curveF.Neg(l2)
-
-	// compute x4 = λ2²-x1-x3
-	x4 := pr.curveF.Eval([][]*baseEl{{l2, l2}, {mone, &p1.X}, {mone, x3}}, []int{1, 1, 1})
-
-	// compute y4 = λ2(x1 - x4)-y1
-	y4 := pr.curveF.Eval([][]*baseEl{{l2, &p1.X}, {mone, l2, x4}, {mone, &p1.Y}}, []int{1, 1, 1})
-
-	p.X = *x4
-	p.Y = *y4
-
-	// compute line2
-	line2.R0 = *l2
-	line2.R1 = *pr.curveF.Mul(l2, &p1.X)
-	line2.R1 = *pr.curveF.Sub(&line2.R1, &p1.Y)
-
-	return &p, &line1, &line2
-}
-
 // doubleStep doubles p1 in affine coordinates, and evaluates the tangent line to p1.
 // https://eprint.iacr.org/2022/1162 (Section 6.1)
 func (pr Pairing) doubleStep(p1 *g2AffP) (*g2AffP, *lineEvaluation) {

[ivokub]

Looks perfect! I also suggested some edits to make the comments a bit more precise and remove unused methods.