Add Circle FRI Section#36
Merged
Merged
Conversation
mellowcroc
reviewed
Oct 14, 2025
|
|
||
| > **Why check proximity to a polynomial?** | ||
| > | ||
| > The prover sends evaluations of a function to the verifier. However, the verifier does not know if this function is a polynomial, or if it satisfies a pre-specified degree bound. FRI allows the verifier to check that the evaluations sent by the prover are indeed "close" to some polynomial of bounded degree. Here, the verifier only checks for proximity; that is, the verifier accepts even if the evaluations sent by the prover are "close enough" to those of a polynomial of bounded degree, they need not be exactly equal. |
Collaborator
There was a problem hiding this comment.
nit) "they need not be exactly equal" -> "thus, they need not be exactly equal"
| $$ | ||
| The prover will find the evaluations of functions $h_{0,0}\;,\; h_{0,1}$ over the domain $I_0$ using the evaluations of $h_0$ over domain $H_0$ as follows: | ||
| $$ | ||
| h_{0,0}(x) = \frac{h_0(x,y)+h_0(x,-y)}{2} \quad, \quad h_{0,1} = \frac{h_0(x,y)-h_0(x,-y)}{2y}. |
Collaborator
There was a problem hiding this comment.
missing
| The prover commits and gives the verifier oracle access to the evaluations $g_0 \in F^{I_0}$. | ||
| - **Round 1**: The prover will decompose the function $h_1$ over the domain $H_1$ into two functions $h_{1,0}\;,\; h_{1,1}$ over the domain $I_1$, same as the decomposition of $h_0$ in **Round 0**. The prover will also decompose the function $g_0$ over domain $I_0$ into two functions $g_{0,0}$ and $g_{0,1}$ over the domain $I_1$ as follows: | ||
| $$ | ||
| g_0(x) = g_{0,0}(2x^2 - 1) + y \cdot g_{0,1}(2x^2 - 1) |
Collaborator
There was a problem hiding this comment.
| $$ | ||
| The prover will find the evaluations of functions $g_{0,0}$ and $g_{0,1}$ over the domain $I_1$ using the evaluations of $g_0$ over domain $I_0$ as follows: | ||
| $$ | ||
| g_{0,0}(2x^2 - 1) = \frac{g_0(x)+g_0(-x)}{2} \quad, \quad g_{0,1}(2x^2 - 1) = \frac{g_0(x)-g_0(-x)}{2y}. |
Collaborator
There was a problem hiding this comment.
| where $g_{i-1,0}, g_{i-1,1} \in F^{I_i}$ are the decomposition of $g_{i-1} \in F^{I_{i-1}}$ and $h_{i,0}, h_{i,1} \in F^{I_i}$ are the decompositions of $h_i \in F^{H_i}$. The prover gives the verifier oracle access to $g_i$, and in the final round $g_r$ is sent in plain. | ||
|
|
||
|
|
||
| 2. _Query Phase_: In the query phase, the verifier checks that the prover computed the folding correctly using the oracles sent by the prover. It consists of $s \geq 1$ rounds. In each round, the verifier samples $P_0 = (x_0,y_0) \in H_0$, and queries the oracles for their values to spot-check the folding identities along the projection trace $P_i = (x_i,y_i) = \pi^i(x_0,y_0)$. That is, for all $i = \{0, 1, 2\}$ the verifier checks that: |
Collaborator
There was a problem hiding this comment.
a different check is done for
Contributor
Author
There was a problem hiding this comment.
This is because of the product with
I have updated the Round 0 of the commit phase to match the folding equation for general
mellowcroc
approved these changes
Oct 16, 2025
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
2 participants
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
No description provided.