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Plonky3 + + + + + + + + + + +
titletags
45. Plonky3 + + + + + + + +
zk-memebasicquick_readPlonky3
+
+ +[Github](https://github.com/ETAAcademy)｜[Twitter](https://twitter.com/ETAAcademy)｜[ETA-ZK-Meme](https://github.com/ETAAcademy/ETAAcademy-ZK-Meme) + +Authors: [Eta](https://twitter.com/pwhattie), looking forward to your joining + +# Plonky3: A Deep Dive into Scalable zk-SNARKs of FRI and UniStark + +**Plonky3** is an advanced zero-knowledge succinct non-interactive argument of knowledge (zk-SNARK) system designed for scalable and efficient cryptographic proof generation. At its core, Plonky3 aims to make zk-SNARKs more scalable by introducing many key components, two of which are **FRI (Fast Reed-Solomon Interactive Oracle Proofs of Proximity)** that streamlines the polynomial commitment process, and **UniStark**, a recursive proving system that allows for faster verification and ensures the scalability of zero-knowledge proofs. + +## 1.FRI (Fast Reed-Solomon Interactive Oracle Proofs of Proximity) in Plonky3 + +In Plonky3, FRI optimizes the commitment scheme using techniques such as polynomial folding, batching, and low-degree extensions (LDE), all of which significantly enhance performance and minimize computation overhead during proof generation. Let's break down the purpose and operations of the main files: + +--- + +- **`two_adic_pcs.rs`**: This file handles polynomials aligned with powers of two. It implements the **Polynomial Commitment Scheme (PCS)** trait, providing the necessary functions for committing, opening, and verifying polynomial data. The main methods—`commit`, `open`, and `verify`—are foundational to the FRI process and allow the prover and verifier to efficiently interact with polynomial commitments. + +- **`proof.rs`**: This file defines the structure of the proof objects generated by the prover during different phases of the FRI protocol, including the **commit phase** and **query-answering phase**. The structures in `proof.rs` are essential for maintaining the integrity of the proof as it is passed between the prover and verifier. + +- **`verifier.rs`**: In the `verifier.rs` file, the most important function here is `verify_query`, which handles the verification of the queries returned by the prover, ensuring that the commitments and proofs align with the expected results. + +- **`fold_even_odd.rs`**: Previously used in Plonky3’s folding process for polynomials, this file has been deprecated in recent versions. + +- **`config.rs`**: This configuration file holds key parameters for the FRI protocol, such as `FriConfig` and `FriGenericConfig`. These configurations include critical values like the **blowup factor** (used in low-degree extensions), the number of queries, and the bits required for proof-of-work. Additionally, `mmcs` (Merkle Matrix Commitment Scheme) is defined here, which is a customized commitment system used for batch polynomial commitments. + +### 1.1 Polynomial Commitment Scheme (PCS) in FRI: Code Structure + +The **Polynomial Commitment Scheme (PCS)** shows the code structure implemented in FRI, which handles matrix folding, polynomial commitments, and verification across multiple rounds, specifically, the folding techniques, Merkle tree generation, and the batch processing that enables FRI's efficiency. + +Before any batch processing occurs, the original data is organized into multiple rounds, where each round consists of matrices with different heights or polynomials of varying degrees. Each round commits into a Merkle tree. Horizontally folding the matrices involves folding matrices of the same height together. For example, when folding a matrix $p(x)$, it requires committing at every fold. After folding, the polynomials of the same degree are reorganized into matrices with a width of two, allowing for simultaneous opening and verification of two values ( i.e., $p_1(-x)$ and $p_1(x)$, one positive and one negative). + +--- + +#### 1.1.1 The `babybear_fri_pcs` and `m31_fri_pcs` Modules + +The `babybear_fri_pcs` module uses the BabyBear domain, while the `m31_fri_pcs` module uses the m31 domain. These modules follow the same structure but operate over different finite fields. + +1. **Selection of types:** The types for the field elements, challenges, and permutations are chosen first. +2. **PCS generation:** The `get_pcs` function generates the PCS (Polynomial Commitment Scheme). +3. **Blowup factors:** Two blowup modules are implemented—`blowup_1` and `blowup_2`—which correspond to blowup factors of 1 and 2, respectively. When the blowup factor is set to 2, the Low Degree Extension (LDE) expands by a factor of 4. Depending on the blowup factor (either 1 or 2), different tests are run using the `make_tests_for_pcs!` macro. + +
Code + +```rust + +mod babybear_fri_pcs { + use super::*; + + type Val = BabyBear; + type Challenge = BinomialExtensionField; + + type Perm = Poseidon2; + type MyHash = PaddingFreeSponge; + type MyCompress = TruncatedPermutation; + + type ValMmcs = + MerkleTreeMmcs<::Packing, ::Packing, MyHash, MyCompress, 8>; + type ChallengeMmcs = ExtensionMmcs; + + type Dft = Radix2DitParallel; + type Challenger = DuplexChallenger; + type MyPcs = TwoAdicFriPcs; + + fn get_pcs(log_blowup: usize) -> (MyPcs, Challenger) { + let perm = Perm::new_from_rng_128( + Poseidon2ExternalMatrixGeneral, + DiffusionMatrixBabyBear::default(), + &mut seeded_rng(), + ); + let hash = MyHash::new(perm.clone()); + let compress = MyCompress::new(perm.clone()); + + let val_mmcs = ValMmcs::new(hash, compress); + let challenge_mmcs = ChallengeMmcs::new(val_mmcs.clone()); + + let fri_config = FriConfig { + log_blowup, + num_queries: 10, + proof_of_work_bits: 8, + mmcs: challenge_mmcs, + }; + + let pcs = MyPcs::new(Dft {}, val_mmcs, fri_config); + (pcs, Challenger::new(perm.clone())) + } + + mod blowup_1 { + make_tests_for_pcs!(super::get_pcs(1)); + } + mod blowup_2 { + make_tests_for_pcs!(super::get_pcs(2)); + } +} + +``` + +
+ +#### 1.1.2 The `macro_rules!` for Generating Matrices + +The `macro_rules!` in `pcs.rs` generates tests dynamically with various matrix dimensions, degrees, and rounds, since PCS tests need to handle various rounds (each containing matrices of different heights) of polynomials efficiently. + +A round here represents a data structure containing matrices of varying heights. In these tests, the log heights are aligned with powers of two to maintain computational efficiency. + +
Code + +```rust + +// Set it up so we create tests inside a module for each pcs, so we get nice error reports +// specific to a failing PCS. + +macro_rules! make_tests_for_pcs { + ($p:expr) => { + #[test] + fn single() { + let p = $p; + for i in 3..6 { + $crate::do_test_fri_pcs(&p, &[&[i]]); + } + } + + #[test] + fn many_equal() { + let p = $p; + for i in 5..8 { + $crate::do_test_fri_pcs(&p, &[&[i; 5]]); + println!("{i} ok"); + } + } + + #[test] + fn many_different() { + let p = $p; + for i in 3..8 { + let degrees = (3..3 + i).collect::>(); + $crate::do_test_fri_pcs(&p, &[°rees]); + } + } + + #[test] + fn many_different_rev() { + let p = $p; + for i in 3..8 { + let degrees = (3..3 + i).rev().collect::>(); + $crate::do_test_fri_pcs(&p, &[°rees]); + } + } + + #[test] + fn multiple_rounds() { + let p = $p; + $crate::do_test_fri_pcs(&p, &[&[3]]); + $crate::do_test_fri_pcs(&p, &[&[3], &[3]]); + $crate::do_test_fri_pcs(&p, &[&[3], &[2]]); + $crate::do_test_fri_pcs(&p, &[&[2], &[3]]); + $crate::do_test_fri_pcs(&p, &[&[3, 4], &[3, 4]]); + $crate::do_test_fri_pcs(&p, &[&[4, 2], &[4, 2]]); + $crate::do_test_fri_pcs(&p, &[&[2, 2], &[3, 3]]); + $crate::do_test_fri_pcs(&p, &[&[3, 3], &[2, 2]]); + $crate::do_test_fri_pcs(&p, &[&[2], &[3, 3]]); + } + }; +} + +``` + +
+ +#### 1.1.3 The `do_test_fri_pcs` Function + +The core function for testing FRI-based PCS is `do_test_fri_pcs`. Let’s break down its key steps: + +1. **Generating domains and polynomials:** Based on the log degrees of each round `log_degrees_by_round`, the `domains_and_polys_by_round` generates polynomials where each log degree dictates the height of the matrix (or the degree of the polynomial). +2. **Committing to polynomials:** Using PCS, the matrices (or polynomials) are committed by `commits_by_round` and supports committing multiple matrices (polynomials) in a single round. +3. **Verifier challenges:** The challenge points `zeta` and `points_by_round` for opening the polynomials are generated, while each column of the matrix can be viewed as a polynomial, and the PCS opens the data by `open(data_and_points, &mut p_challenger)`. +4. **Verification:** Finally, the proof is verified, ensuring that the commitments made by the prover match the verifier’s expectations by `pcs.verify`. + +
Code + +```rust + +fn do_test_fri_pcs( + (pcs, challenger): &(P, Challenger), + log_degrees_by_round: &[&[usize]], +) where + P: Pcs, + P::Domain: PolynomialSpace, + Val: Field, + Standard: Distribution, + Challenge: ExtensionField, + Challenger: Clone + CanObserve + FieldChallenger, +{ + let num_rounds = log_degrees_by_round.len(); + let mut rng = seeded_rng(); + + let mut p_challenger = challenger.clone(); + + let domains_and_polys_by_round = log_degrees_by_round + .iter() + .map(|log_degrees| { + log_degrees + .iter() + .map(|&log_degree| { + let d = 1 << log_degree; + // random width 5-15 + let width = 5 + rng.gen_range(0..=10); + ( + pcs.natural_domain_for_degree(d), + RowMajorMatrix::::rand(&mut rng, d, width), + ) + }) + .collect_vec() + }) + .collect_vec(); + + let (commits_by_round, data_by_round): (Vec<_>, Vec<_>) = domains_and_polys_by_round + .iter() + .map(|domains_and_polys| pcs.commit(domains_and_polys.clone())) + .unzip(); + assert_eq!(commits_by_round.len(), num_rounds); + assert_eq!(data_by_round.len(), num_rounds); + p_challenger.observe_slice(&commits_by_round); + + let zeta: Challenge = p_challenger.sample_ext_element(); + + let points_by_round = log_degrees_by_round + .iter() + .map(|log_degrees| vec![vec![zeta]; log_degrees.len()]) + .collect_vec(); + let data_and_points = data_by_round.iter().zip(points_by_round).collect(); + let (opening_by_round, proof) = pcs.open(data_and_points, &mut p_challenger); + assert_eq!(opening_by_round.len(), num_rounds); + + // Verify the proof. + let mut v_challenger = challenger.clone(); + v_challenger.observe_slice(&commits_by_round); + let verifier_zeta: Challenge = v_challenger.sample_ext_element(); + assert_eq!(verifier_zeta, zeta); + + let commits_and_claims_by_round = izip!( + commits_by_round, + domains_and_polys_by_round, + opening_by_round + ) + .map(|(commit, domains_and_polys, openings)| { + let claims = domains_and_polys + .iter() + .zip(openings) + .map(|((domain, _), mat_openings)| (*domain, vec![(zeta, mat_openings[0].clone())])) + .collect_vec(); + (commit, claims) + }) + .collect_vec(); + assert_eq!(commits_and_claims_by_round.len(), num_rounds); + + pcs.verify(commits_and_claims_by_round, &proof, &mut v_challenger) + .unwrap() +} + +``` + +
+ +### 1.2 Detailed Functionality of `two_adic_pcs.rs` + +The `two_adic_pcs.rs` module implements a Polynomial Commitment Scheme (PCS) tailored to operate over a power-of-two aligned domain. This alignment allows for optimized low-degree extensions, bit-reversal ordering, and efficient matrix handling in a two-adic setting. + +The three core functionalities provided by this PCS implementation are encapsulated in the `PCS` trait: `commit`, `open`, and `verify`. Each of these methods contributes to various stages of polynomial handling, from committing polynomials to verifying their evaluations. + +--- + +#### 1.2.1: `commit` Function + +The `commit` function is responsible for generating a cryptographic commitment to a set of polynomials, encapsulated in matrices. Each matrix is paired with a corresponding domain, and the goal is to ensure that the commitments are aligned with the domain size. + +##### Key Steps: + +1. **Input Arguments**: The function accepts `evaluations`, which are a collection of matrix-domain pairs. Each matrix represents evaluations of a polynomial, and the domain determines the degree and size constraints of that matrix. +2. **Domain-Size Consistency**: It first checks if the height of each matrix corresponds to the size of the domain, ensuring that the number of evaluations matches the expected degree constraints. + +3. **Low-Degree Extension (LDE)**: For each matrix, the function computes the **low-degree extension** (LDE) over a coset using `self.dft.coset_lde_batch`. This extension maps the matrix into a higher-dimensional space, enabling robust commitments even when the degree is extended. + +4. **Bit-Reversing**: After LDE, the rows of each matrix are bit-reversed using `bit_reverse_rows`, which arranges evaluations at $x$ and $-x$ in an alternating fashion. This rearrangement is crucial for efficient evaluation in later stages. + +5. **Commitment Using MMCs**: The resulting matrices, post-extension and bit-reversal, are committed using multi-message commitments (MMCs), producing a cryptographic commitment that serves as the public representation of the polynomial. + +
Code + +```rust + + fn commit( + &self, + evaluations: Vec<(Self::Domain, RowMajorMatrix)>, + ) -> (Self::Commitment, Self::ProverData) { + let ldes: Vec<_> = evaluations + .into_iter() + .map(|(domain, evals)| { + assert_eq!(domain.size(), evals.height()); + let shift = Val::generator() / domain.shift; + // Commit to the bit-reversed LDE. + self.dft + .coset_lde_batch(evals, self.fri.log_blowup, shift) + .bit_reverse_rows() + .to_row_major_matrix() + }) + .collect(); + + self.mmcs.commit(ldes) + } + +``` + +
+ +#### 1.2.2: `open` Function + +The `open` function plays a pivotal role in computing **quotient polynomials**. Given a set of polynomials and evaluation points, it produces the batched quotient polynomial values for a verifier to check. The quotient polynomial $q(x)$ is computed as: + +$$ +q(x_i) = \sum_{k=0}^{w_l-1} \alpha^k \cdot \frac{f_k(x_i) - y_k}{x_i - z} +$$ + +Where: + +- $f_k(x_i)$ represents the $k$-th polynomial evaluated at $x_i$. +- $y_k = f_k(z)$ is the value of $f_k$ at the opening point $z$. +- $\alpha$ is a random challenge used to batch multiple evaluations into a single quotient. + +##### Optimizations for Efficiency: + +1. **Pre-computation of Inverses**: To avoid repeated calculations, the function precomputes $\frac{1}{x_i - z}$ over the largest domain. These precomputed values are reused across smaller domains by simple truncation, leveraging the power-of-two structure of the domain sizes. + +2. **Batching of Alpha Powers**: Instead of recalculating powers of $\alpha$ for each polynomial individually, these powers are computed once for the largest matrix width and then reused with an offset for subsequent computations. This reduces the computational overhead of extension field multiplications. + +3. **Combination of Alpha and Evaluation Values**: For each matrix, the function accumulates the quotient polynomial values by combining the precomputed alpha powers, evaluation values $f_k(x_i)$, and the precomputed inverses. This batching process is optimized to minimize expensive extension field multiplications. + +
Code + +```rust + + fn open( + &self, + // For each round, + rounds: Vec<( + &Self::ProverData, + // for each matrix, + Vec< + // points to open + Vec, + >, + )>, + challenger: &mut Challenger, + ) -> (OpenedValues, Self::Proof) { + /* + + A quick rundown of the optimizations in this function: + We are trying to compute sum_i alpha^i * (p(X) - y)/(X - z), + for each z an opening point, y = p(z). Each p(X) is given as evaluations in bit-reversed order + in the columns of the matrices. y is computed by barycentric interpolation. + X and p(X) are in the base field; alpha, y and z are in the extension. + The primary goal is to minimize extension multiplications. + + - Instead of computing all alpha^i, we just compute alpha^i for i up to the largest width + of a matrix, then multiply by an "alpha offset" when accumulating. + a^0 x0 + a^1 x1 + a^2 x2 + a^3 x3 + ... + = a^0 ( a^0 x0 + a^1 x1 ) + a^2 ( a^0 x2 + a^1 x3 ) + ... + (see `alpha_pows`, `alpha_pow_offset`, `num_reduced`) + + - For each unique point z, we precompute 1/(X-z) for the largest subgroup opened at this point. + Since we compute it in bit-reversed order, smaller subgroups can simply truncate the vector. + (see `inv_denoms`) + + - Then, for each matrix (with columns p_i) and opening point z, we want: + for each row (corresponding to subgroup element X): + reduced[X] += alpha_offset * sum_i [ alpha^i * inv_denom[X] * (p_i[X] - y[i]) ] + + We can factor out inv_denom, and expand what's left: + reduced[X] += alpha_offset * inv_denom[X] * sum_i [ alpha^i * p_i[X] - alpha^i * y[i] ] + + And separate the sum: + reduced[X] += alpha_offset * inv_denom[X] * [ sum_i [ alpha^i * p_i[X] ] - sum_i [ alpha^i * y[i] ] ] + + And now the last sum doesn't depend on X, so we can precompute that for the matrix, too. + So the hot loop (that depends on both X and i) is just: + sum_i [ alpha^i * p_i[X] ] + + with alpha^i an extension, p_i[X] a base + + */ + + // Batch combination challenge + let alpha: Challenge = challenger.sample_ext_element(); + + let mats_and_points = rounds + .iter() + .map(|(data, points)| { + ( + self.mmcs + .get_matrices(data) + .into_iter() + .map(|m| m.as_view()) + .collect_vec(), + points, + ) + }) + .collect_vec(); + let mats = mats_and_points + .iter() + .flat_map(|(mats, _)| mats) + .collect_vec(); + + let global_max_height = mats.iter().map(|m| m.height()).max().unwrap(); + let log_global_max_height = log2_strict_usize(global_max_height); + + // For each unique opening point z, we will find the largest degree bound + // for that point, and precompute 1/(X - z) for the largest subgroup (in bitrev order). + let inv_denoms = compute_inverse_denominators(&mats_and_points, Val::generator()); + + let mut all_opened_values: OpenedValues = vec![]; + + let mut reduced_openings: [_; 32] = core::array::from_fn(|_| None); + let mut num_reduced = [0; 32]; + + for (mats, points) in mats_and_points { + let opened_values_for_round = all_opened_values.pushed_mut(vec![]); + for (mat, points_for_mat) in izip!(mats, points) { + let log_height = log2_strict_usize(mat.height()); + let reduced_opening_for_log_height = reduced_openings[log_height] + .get_or_insert_with(|| vec![Challenge::zero(); mat.height()]); + debug_assert_eq!(reduced_opening_for_log_height.len(), mat.height()); + + let opened_values_for_mat = opened_values_for_round.pushed_mut(vec![]); + for &point in points_for_mat { + let _guard = + info_span!("reduce matrix quotient", dims = %mat.dimensions()).entered(); + + // Use Barycentric interpolation to evaluate the matrix at the given point. + let ys = info_span!("compute opened values with Lagrange interpolation") + .in_scope(|| { + let (low_coset, _) = + mat.split_rows(mat.height() >> self.fri.log_blowup); + interpolate_coset( + &BitReversalPerm::new_view(low_coset), + Val::generator(), + point, + ) + }); + + let alpha_pow_offset = alpha.exp_u64(num_reduced[log_height] as u64); + let reduced_ys: Challenge = dot_product(alpha.powers(), ys.iter().copied()); + + info_span!("reduce rows").in_scope(|| { + mat.dot_ext_powers(alpha) + .zip(reduced_opening_for_log_height.par_iter_mut()) + // This might be longer, but zip will truncate to smaller subgroup + // (which is ok because it's bitrev) + .zip(inv_denoms.get(&point).unwrap().par_iter()) + .for_each(|((reduced_row, ro), &inv_denom)| { + *ro += alpha_pow_offset * (reduced_row - reduced_ys) * inv_denom + }) + }); + + num_reduced[log_height] += mat.width(); + opened_values_for_mat.push(ys); + } + } + } + + let fri_input = reduced_openings.into_iter().rev().flatten().collect_vec(); + + let g: TwoAdicFriGenericConfigForMmcs = + TwoAdicFriGenericConfig(PhantomData); + + let fri_proof = prover::prove(&g, &self.fri, fri_input, challenger, |index| { + rounds + .iter() + .map(|(data, _)| { + let log_max_height = log2_strict_usize(self.mmcs.get_max_height(data)); + let bits_reduced = log_global_max_height - log_max_height; + let reduced_index = index >> bits_reduced; + let (opened_values, opening_proof) = self.mmcs.open_batch(reduced_index, data); + BatchOpening { + opened_values, + opening_proof, + } + }) + .collect() + }); + + (all_opened_values, fri_proof) + } + +``` + +
+ +#### 1.2.3: `verify` Function + +The `verify` function ensures that the polynomial evaluations committed to by the prover are valid by checking the provided proof against the original commitment. + +##### Steps: + +1. **Initialization of Reduced Openings**: A BTreeMap is initialized to store batched quotient polynomial openings, keyed by the logarithmic height of each matrix. + +The first step in handling FRI proofs involves setting up a data structure to manage the polynomial opening data. A mutable `BTreeMap` is initialized to store tuples keyed by the `log_height` of each matrix. These tuples will contain: + +- The accumulated batched parameter, represented as powers of alpha. +- The accumulated batched quotient polynomial opened values. + +This structure will be updated iteratively through the processing of multiple rounds of polynomial data. The core logic for calculating the quotient polynomials’ openings across all rounds follows the formula: + +$$ +\{q_d(x)\}_{d\in S}=\left\{\sum_{k}\sum_{j}\alpha^{k+j } \cdot \frac{p_{jk}(x) - y_{jk}}{x-z_k} \right\}_{d\in S} +$$ + +Here, the indices are defined as: + +- $k \in [1,t]$, where $t$ is the number of open points. +- $j \in [1,m]$, where $m$ is the number of polynomials. +- $d$ is the degree of the polynomial, $p_{jk}$, which is opened at the $k_{th}$ open point with a degree $d$. + +This calculation is crucial for batching the quotient polynomials' opened values across multiple rounds. + +2. **Iterating Over Rounds**: The function iterates over each round’s proof and commitment data, verifying the batched quotient polynomial openings. For each matrix in the round, the verifier checks that the evaluation of the polynomial at a given point matches the values provided by the prover. + +3. **Closure for Matrix Verification**: The key logic for verification is encapsulated in a closure that iterates over all matrices and computes the accumulated quotient polynomial values. + +
Code + +```rust + fn verify( + &self, + // For each round: + rounds: Vec<( + Self::Commitment, + // for each matrix: + Vec<( + // its domain, + Self::Domain, + // for each point: + Vec<( + // the point, + Challenge, + // values at the point + Vec, + )>, + )>, + )>, + proof: &Self::Proof, + challenger: &mut Challenger, + ) -> Result<(), Self::Error> { + // Batch combination challenge + let alpha: Challenge = challenger.sample_ext_element(); + + let log_global_max_height = proof.commit_phase_commits.len() + self.fri.log_blowup; + + let g: TwoAdicFriGenericConfigForMmcs = + TwoAdicFriGenericConfig(PhantomData); + + verifier::verify(&g, &self.fri, proof, challenger, |index, input_proof| { + // TODO: separate this out into functions + + // log_height -> (alpha_pow, reduced_opening) + let mut reduced_openings = BTreeMap::::new(); + + for (batch_opening, (batch_commit, mats)) in izip!(input_proof, &rounds) { + let batch_heights = mats + .iter() + .map(|(domain, _)| domain.size() << self.fri.log_blowup) + .collect_vec(); + let batch_dims = batch_heights + .iter() + // TODO: MMCS doesn't really need width; we put 0 for now. + .map(|&height| Dimensions { width: 0, height }) + .collect_vec(); + + let batch_max_height = batch_heights.iter().max().expect("Empty batch?"); + let log_batch_max_height = log2_strict_usize(*batch_max_height); + let bits_reduced = log_global_max_height - log_batch_max_height; + let reduced_index = index >> bits_reduced; + + self.mmcs.verify_batch( + batch_commit, + &batch_dims, + reduced_index, + &batch_opening.opened_values, + &batch_opening.opening_proof, + )?; + for (mat_opening, (mat_domain, mat_points_and_values)) in + izip!(&batch_opening.opened_values, mats) + { + let log_height = log2_strict_usize(mat_domain.size()) + self.fri.log_blowup; + + let bits_reduced = log_global_max_height - log_height; + let rev_reduced_index = reverse_bits_len(index >> bits_reduced, log_height); + + // todo: this can be nicer with domain methods? + + let x = Val::generator() + * Val::two_adic_generator(log_height).exp_u64(rev_reduced_index as u64); + + let (alpha_pow, ro) = reduced_openings + .entry(log_height) + .or_insert((Challenge::one(), Challenge::zero())); + + for (z, ps_at_z) in mat_points_and_values { + for (&p_at_x, &p_at_z) in izip!(mat_opening, ps_at_z) { + let quotient = (-p_at_z + p_at_x) / (-*z + x); + *ro += *alpha_pow * quotient; + *alpha_pow *= alpha; + } + } + } + } + + // Return reduced openings descending by log_height. + Ok(reduced_openings + .into_iter() + .rev() + .map(|(log_height, (_alpha_pow, ro))| (log_height, ro)) + .collect()) + }) + .expect("fri err"); + + Ok(()) + } +} + +``` + +
+ +### 1.3 Vertical Folding in the FRI Protocol: A Prover and Verifier Perspective + +In the Fast Reed-Solomon Interactive Oracle Proof of Proximity (FRI) protocol, both the prover and verifier must handle numerous polynomials, their evaluations, and corresponding proofs. One of the key processes within this interaction is **vertical folding**, which plays a crucial role in the efficient proof of polynomial degrees and the batching of quotient polynomials. + +--- + +#### 1.3.1 Prover’s Input, Folding and Batching + +The prover in FRI handles a batch of polynomials and needs to evaluate them over specific points, compute quotient polynomials, and perform folding. The prover's input consists of the following elements: + +1. **Inputs**: The prover starts with a batch of polynomials $\{p_j(X)\}_{j=1}^{m}$, their evaluations after a low-degree extension (LDE), and a set of open points $\{z_{jk}\}_{k=0}^{t_j}$. The prover commits to these evaluated polynomials in each round $\{Commit_r\}_{r\in \text{all rounds}}$. + +2. **Batched Quotient Polynomial**: For each distinct degree $d$, the prover computes a batched quotient polynomial as follows: + + $$ + q)d(\omega^i) = \sum_{j, k} \alpha^{k+j} \cdot \frac{p_j(\omega^i) - y_{jk}}{\omega^i - z_k} + $$ + + This results in a set of batched quotient polynomials for all degrees $d \in S$. + +3. **Folding and Batching**: + - **Initialization**: Set $f = q_{dm}$, where $dm$ is the highest degree. + - **Commitment**: Use MMCS to commit $f$ as a matrix. + - **Folding**: Fold $f$ as $f = f_e + \beta f_o$, where $f_e$ and $f_o$ are the even and odd parts of the polynomial, and $\beta$ is a random challenge. + - **Batching**: Update $f$ by adding the next quotient polynomial, $q_d$. + - **Repeat**: Continue folding and batching until $f$ becomes a constant polynomial. + +At the end of the process, the prover has: + +- Commitments to intermediate polynomials $\{C_{f_d}\}_{d \in S}$. +- A final constant polynomial $f_c$. + +This prepares the prover to answer the verifier's queries efficiently, providing opening proofs for verification. + +
Code + +```rust +#[instrument(name = "commit phase", skip_all)] +fn commit_phase( + g: &G, + config: &FriConfig, + inputs: Vec>, + challenger: &mut Challenger, +) -> CommitPhaseResult +where + Val: Field, + Challenge: ExtensionField, + M: Mmcs, + Challenger: FieldChallenger + CanObserve, + G: FriGenericConfig, +{ + let mut inputs_iter = inputs.into_iter().peekable(); + let mut folded = inputs_iter.next().unwrap(); + let mut commits = vec![]; + let mut data = vec![]; + + while folded.len() > config.blowup() { + let leaves = RowMajorMatrix::new(folded, 2); + let (commit, prover_data) = config.mmcs.commit_matrix(leaves); + challenger.observe(commit.clone()); + + let beta: Challenge = challenger.sample_ext_element(); + // We passed ownership of `current` to the MMCS, so get a reference to it + let leaves = config.mmcs.get_matrices(&prover_data).pop().unwrap(); + folded = g.fold_matrix(beta, leaves.as_view()); + + commits.push(commit); + data.push(prover_data); + + if let Some(v) = inputs_iter.next_if(|v| v.len() == folded.len()) { + izip!(&mut folded, v).for_each(|(c, x)| *c += x); + } + } + + // We should be left with `blowup` evaluations of a constant polynomial. + assert_eq!(folded.len(), config.blowup()); + let final_poly = folded[0]; + for x in folded { + assert_eq!(x, final_poly); + } + challenger.observe_ext_element(final_poly); + + CommitPhaseResult { + commits, + data, + final_poly, + } +} + +``` + +
+ +#### 1.3.2 Vertical Folding in the Verifier + +The verifier's task is to validate the prover's folded polynomials and opening proofs. The vertical folding process for the verifier follows these steps: + +1. **Verifier Input**: The verifier receives the prover’s commitments $\{Commit_r\}_{r \in \text{all rounds}}$, all input and query opened values, and the sample query index $\delta$. + +2. **Initialize Folding**: The verifier initializes the folded evaluation $e = 0$. + +3. **Batching**: The verifier batches the folded evaluation by setting $e = e + q_d(\delta)$ for each degree $d$, ensuring the degree of $q_d(\delta)$ matches the current degree of $e$. + +4. **Check Openings**: The verifier checks the opening of $e$ by comparing the computed value with the prover's opening proof. + +5. **Folding**: The verifier computes the next folded evaluation using the current $e$, the folding parameter $\beta$, the query index $\delta$, and the sibling value corresponding to the current fold. + +6. **Iteration**: Steps 2–5 are repeated for the number of iterations corresponding to the logarithm of the degree. + +Finally, the verifier arrives at a final folded evaluation $e\_{\text{final}}$ and checks its equality with the constant polynomial $f_c$ sent by the prover. + +
Code + +```rust + +fn verify_query<'a, G, F, M>( + g: &G, + config: &FriConfig, + mut index: usize, + steps: impl Iterator>, + reduced_openings: Vec<(usize, F)>, + log_max_height: usize, +) -> Result> +where + F: Field, + M: Mmcs + 'a, + G: FriGenericConfig, +{ + let mut folded_eval = F::zero(); + let mut ro_iter = reduced_openings.into_iter().peekable(); + + for (log_folded_height, (&beta, comm, opening)) in izip!((0..log_max_height).rev(), steps) { + if let Some((_, ro)) = ro_iter.next_if(|(lh, _)| *lh == log_folded_height + 1) { + folded_eval += ro; + } + + let index_sibling = index ^ 1; + let index_pair = index >> 1; + + let mut evals = vec![folded_eval; 2]; + evals[index_sibling % 2] = opening.sibling_value; + + let dims = &[Dimensions { + width: 2, + height: 1 << log_folded_height, + }]; + config + .mmcs + .verify_batch( + comm, + dims, + index_pair, + &[evals.clone()], + &opening.opening_proof, + ) + .map_err(FriError::CommitPhaseMmcsError)?; + + index = index_pair; + + folded_eval = g.fold_row(index, log_folded_height, beta, evals.into_iter()); + } + + debug_assert!(index < config.blowup(), "index was {}", index); + debug_assert!( + ro_iter.next().is_none(), + "verifier reduced_openings were not in descending order?" + ); + + Ok(folded_eval) +} + +``` + +
+ +## 2.Uni-STARK: Overview of Prover and Verifier Roles in Plonky3 + +The Uni-STARK system in **Plonky3** generally follows the structure of STARKs as mentioned before. Below is a step-by-step breakdown of the process, which includes committing to trace and quotient polynomials, opening them for verification, and the eventual query phase for proof validation. + +### 2.1. Commit Phase + +The **commit** phase involves committing to both the trace polynomials and the quotient polynomials: + +--- + +**Step 1: Commit to Trace Polynomials** + +- First, we calculate the trace, which could be something like a Fibonacci sequence. The trace table consists of several columns, each representing a trace polynomial. After performing a low-degree extension (LDE), these trace polynomials are committed to. + - The trace table is represented as a `RowMajorMatrix>`. The trace polynomials are committed over a multiplicative subgroup called the trace domain. To prevent the quotient polynomial from being computed over the same domain (which would result in the quotient being zero), the `pcs.commit` function offsets the trace domain to a coset and performs the commitment over the LDE combined with a bit-reversal technique. + +**Step 2: Commit to the Quotient Polynomial** + +- The quotient polynomial is computed by applying the constraints to the trace polynomials (like Fibonacci recurrence rules) and then dividing the result by a zero polynomial $Z(X)$ to form the quotient polynomial. + - A coset corresponding to the trace domain is selected, with the size of the coset group being $\log(\text{degree}) + \log(\text{quotient degree})$. + - The trace on the quotient domain represents the trace polynomials on that domain. Transitioning from the trace domain to the quotient domain involves shifting by the coset's offset. For example, converting a polynomial from the original domain to the coset involves a shift $x \mapsto gx$, where $g$ is the coset generator. + - Quotient values are calculated, and the degrees of the quotient polynomials are aligned using a random $\alpha$. In Plonky3, the degree of the trace polynomial is usually less than the quotient polynomial degree, but here the quotient polynomial degree equals $\log(\text{degree}) + \log(\text{quotient degree})$. + - Using the `get_evaluations_on_domain` function from `two_adic_pcs.rs`, LDE evaluations are performed. The number of rows needed is adjusted based on the quotient domain, and bit-reversed rows are processed from the Merkle tree or from previous trace commitments to match the larger quotient degree. + +Key formulas include: + +- $C_0(X) = a(X) + b(X) - b(gX)$ (enforcing the first transition constraint) +- $C_1(X) = a(gX) - b(X)$ (enforcing the second transition constraint) +- $q_0(X) = \frac{C_0(X)}{Z_H(X)}$ and $q_1(X) = \frac{C_1(X)}{Z_H(X)}$ + +To align degrees, a random $\alpha$ is used, producing modified quotient polynomials $q_0'(X)$ and $q_1'(X)$. These are combined to form $Q(X) = q_0'(X) + q_1'(X)$, where the domain is on a coset. + +**Step 3: Splitting the Quotient Polynomial into Chunks** + +- To optimize the FRI (Fast Reed-Solomon Interactive Oracle Proof of Proximity) protocol, the quotient polynomial is split into chunks. These smaller quotient polynomials are of the same degree as the trace polynomial. + - For example, suppose we have a multiplicative subgroup of order 6, and we want to split the quotient polynomial into two smaller polynomials. This splitting is done by evaluating $Q(X)$ at generators $w^2$ and $w^4$, where $w$ is a two-adic generator. + - The quotient polynomial is split as $Q(z) = Q_1(z^S) + z \cdot Q_2(z^S)$, where evaluations take place at domain generators shifted by cosets. + +After splitting, each chunk is committed individually as follows: - $Q_1(X)$ and $Q_2(X)$ are committed using their low-degree extensions. - A Merkle tree is constructed from the commitment matrices of the quotient polynomial chunks, and the final commitment is recorded. + +In the Fiat–Shamir phase, random challenge points $\zeta$ and $\zeta_{\text{next}}$ are used to verify that the committed trace and quotient polynomials satisfy the required constraint relationships. The `pcs.open` function retrieves the necessary proofs for verification, ensuring consistency between the commitments and the actual trace polynomials. + +
Code + +```rust + + let degree = trace.height(); + let log_degree = log2_strict_usize(degree); + + let log_quotient_degree = get_log_quotient_degree::, A>(air, 0, public_values.len()); + let quotient_degree = 1 << log_quotient_degree; + + let pcs = config.pcs(); + let trace_domain = pcs.natural_domain_for_degree(degree); + + let (trace_commit, trace_data) = + info_span!("commit to trace data").in_scope(|| pcs.commit(vec![(trace_domain, trace)])); + + // Observe the instance. + challenger.observe(Val::::from_canonical_usize(log_degree)); + // TODO: Might be best practice to include other instance data here; see verifier comment. + + challenger.observe(trace_commit.clone()); + challenger.observe_slice(public_values); + let alpha: SC::Challenge = challenger.sample_ext_element(); + + let quotient_domain = + trace_domain.create_disjoint_domain(1 << (log_degree + log_quotient_degree)); + + let trace_on_quotient_domain = pcs.get_evaluations_on_domain(&trace_data, 0, quotient_domain); + + let quotient_values = quotient_values( + air, + public_values, + trace_domain, + quotient_domain, + trace_on_quotient_domain, + alpha, + ); + let quotient_flat = RowMajorMatrix::new_col(quotient_values).flatten_to_base(); + let quotient_chunks = quotient_domain.split_evals(quotient_degree, quotient_flat); + let qc_domains = quotient_domain.split_domains(quotient_degree); + + let (quotient_commit, quotient_data) = info_span!("commit to quotient poly chunks") + .in_scope(|| pcs.commit(izip!(qc_domains, quotient_chunks).collect_vec())); + challenger.observe(quotient_commit.clone()); + + let commitments = Commitments { + trace: trace_commit, + quotient_chunks: quotient_commit, + }; + + let zeta: SC::Challenge = challenger.sample(); + let zeta_next = trace_domain.next_point(zeta).unwrap(); + +``` + +
+ +### 2.2 Opening Phase + +--- + +The **opening** phase allows the prover to reveal specific evaluations of the trace and quotient polynomials to the verifier: + +#### 2.2.1 Constructing the Opening Proof + +When the verifier receives the proof, it needs to verify that the trace and quotient polynomials satisfy the constraints at random points, typically denoted as $\zeta$ and $\zeta\_{\text{next}}$, which are sampled in an extension field. Since the committed trace polynomials are over the base field, $\zeta$ does not belong to the original multiplicative subgroup over which the trace polynomials were committed. + +**Step 1: Opening the Trace Polynomial** + +- To ensure that the trace polynomial satisfies the given constraints, we need to "open" the trace polynomial at the points $\zeta$ and $\zeta\_{\text{next}}$. +- Since the trace polynomials at $\zeta$ and $\zeta\_{\text{next}}$ have the same degree, we can linearly combine them into a reduced polynomial, $\text{reduce}\_0(X)$, which simplifies further computations. + +**Step 2: Opening the Quotient Polynomial** + +- Similarly, we open the quotient polynomial at $\zeta$. Here, we compute two quotient-related polynomials, $ldt_2(X)$ and $ldt_3(X)$, which represent quotient polynomials over different chunks (since the quotient polynomial has been split into chunks). +- The quotient chunks at $\zeta$ are combined into a reduced polynomial $\text{reduce}\_1(X)$. This polynomial also has the same degree as $\text{reduce}\_0(X)$, allowing the two reduced polynomials to be tested together. + +#### 2.2.2. Low-Degree Testing of the Reduced Polynomials + +Once the reduced polynomials $\text{reduce}\_0(X)$ and $\text{reduce}\_1(X)$ are constructed, they undergo a **low-degree test** to verify that both polynomials have a degree lower than a specified threshold. + +**Steps for Low-Degree Testing**: + +1. **Opening the Trace Polynomial**: + - Open the trace polynomial at points $\zeta$ and $\zeta\_{\text{next}}$, as well as the quotient polynomial chunks $Q_1(X)$ and $Q_2(X)$ at $\zeta$. + - These openings yield intermediate values used to compute the reduced polynomials. +2. **Constructing the Reduced Polynomial**: + - Compute the reduced trace polynomial: + \[ + \text{reduce}_0(X) = \sum \alpha^i \cdot ldt_0(X)\_i + \alpha^{i+1} \cdot ldt_0'(X) + \] + where $ldt_0(X)$ and $ldt_0'(X)$ are the Lagrange polynomials formed from the trace evaluations at $\zeta$ and $\zeta_{\text{next}}$. + - Similarly, compute the reduced quotient polynomial: + \[ + \text{reduce}\_1(X) = \sum \alpha^i \cdot ldt_2(X)\_i + \alpha^{i+1} \cdot ldt_3(X) + \] + where $ldt_2(X)$ and $ldt_3(X)$ are Lagrange polynomials derived from quotient evaluations. +3. **Performing the Low-Degree Test**: + - Both reduced polynomials $\text{reduce}\_0(X)$ and $\text{reduce}\_1(X)$ undergo a low-degree test to verify their degree bounds. + +
Code + +```rust + + let (opened_values, opening_proof) = info_span!("open").in_scope(|| { + pcs.open( + vec![ + (&trace_data, vec![vec![zeta, zeta_next]]), + ( + "ient_data, + // open every chunk at zeta + (0..quotient_degree).map(|_| vec![zeta]).collect_vec(), + ), + ], + challenger, + ) + }); + let trace_local = opened_values[0][0][0].clone(); + let trace_next = opened_values[0][0][1].clone(); + let quotient_chunks = opened_values[1].iter().map(|v| v[0].clone()).collect_vec(); + let opened_values = OpenedValues { + trace_local, + trace_next, + quotient_chunks, + }; + Proof { + commitments, + opened_values, + opening_proof, + degree_bits: log_degree, + } + +``` + +
+ +### 2.3. Query Phase: Verifying Constraints + +In the query phase, the verifier confirms that the trace polynomial at $\zeta$ and $\zeta\_{\text{next}}$, along with the quotient polynomial, satisfy the constraints. This involves several steps: + +**Step 1: Verifying $Q(\zeta)$** + +- The verifier checks if the trace polynomial at $\zeta$ and $\zeta\_{\text{next}}$ satisfies the constraints by applying the transition rules, allowing the calculation of $Q(\zeta)$. +- The quotient polynomial values $Q_1(\zeta)$ and $Q_2(\zeta)$ are combined to reconstruct $Q(\zeta)$. The verifier then compares the reconstructed $Q(\zeta)$ to the expected result. If they match, the constraints are satisfied. +- In the code, this is done by folding the constraints and comparing the results: + +```rust +if folded_constraints = sels.inv_zeroifier != quotient { ... } +``` + +**Step 3.2: Addressing ZPS and Domain Shifts** + +- The process of reconstructing $Q(\zeta)$ introduces a potential **Zero-Polynomial Shift (ZPS)** problem. When using Lagrange interpolation to reconstruct $Q(\zeta)$, errors can occur due to domain shrinking. +- For example, $Q(X) = Q_1(X^2) + X \cdot Q_1(-X^2)$ has a smaller domain, so additional corrections are needed to account for the missing elements from the other coset domains. This is handled by computing a ZPS correction factor to adjust for these domain shifts. + +The Lagrange interpolants are adjusted as follows: + +$$ +zps[i] = \prod_{j \neq i} \frac{\left(\frac{x}{\text{shift}_j}\right)^n - 1}{\left(\frac{\text{shift}_i}{\text{shift}\_j}\right)^n - 1} +$$ + +The verifier then uses this ZPS correction to verify the consistency of the trace and quotient evaluations at $\zeta$ and $\zeta\_{\text{next}}$, ensuring that all domain shifts are correctly handled. + +
Code + +```rust + +pcs.verify( + vec![ + ( + commitments.trace.clone(), + vec![( + trace_domain, + vec![ + (zeta, opened_values.trace_local.clone()), + (zeta_next, opened_values.trace_next.clone()), + ], + )], + ), + ( + commitments.quotient_chunks.clone(), + quotient_chunks_domains + .iter() + .zip(&opened_values.quotient_chunks) + .map(|(domain, values)| (*domain, vec![(zeta, values.clone())])) + .collect_vec(), + ), + ], + opening_proof, + challenger, + ) + .map_err(VerificationError::InvalidOpeningArgument)?; + + let zps = quotient_chunks_domains + .iter() + .enumerate() + .map(|(i, domain)| { + quotient_chunks_domains + .iter() + .enumerate() + .filter(|(j, _)| *j != i) + .map(|(_, other_domain)| { + other_domain.zp_at_point(zeta) + * other_domain.zp_at_point(domain.first_point()).inverse() + }) + .product::() + }) + .collect_vec(); + + let quotient = opened_values + .quotient_chunks + .iter() + .enumerate() + .map(|(ch_i, ch)| { + ch.iter() + .enumerate() + .map(|(e_i, &c)| zps[ch_i] * SC::Challenge::monomial(e_i) * c) + .sum::() + }) + .sum::(); + + let sels = trace_domain.selectors_at_point(zeta); + + let main = VerticalPair::new( + RowMajorMatrixView::new_row(&opened_values.trace_local), + RowMajorMatrixView::new_row(&opened_values.trace_next), + ); + + let mut folder = VerifierConstraintFolder { + main, + public_values, + is_first_row: sels.is_first_row, + is_last_row: sels.is_last_row, + is_transition: sels.is_transition, + alpha, + accumulator: SC::Challenge::zero(), + }; + air.eval(&mut folder); + let folded_constraints = folder.accumulator; + + // Finally, check that + // folded_constraints(zeta) / Z_H(zeta) = quotient(zeta) + if folded_constraints * sels.inv_zeroifier != quotient { + return Err(VerificationError::OodEvaluationMismatch); + } + + +``` + +
+ +### Conclusion + +As demand for zk-SNARKs continues to grow, Plonky3's blend of FRI and UniStark technologies positions it as a cutting-edge tool for scaling cryptographic proofs, enabling faster, more secure, and more efficient decentralized ecosystems. + +
+ Image 1 +
diff --git a/Appendix/Plonky3/.github/dependabot.yml b/Appendix/Plonky3/.github/dependabot.yml new file mode 100644 index 0000000..53f8242 --- /dev/null +++ b/Appendix/Plonky3/.github/dependabot.yml @@ -0,0 +1,6 @@ +version: 2 +updates: + - package-ecosystem: "cargo" + directory: "/" + schedule: + interval: "weekly" diff --git a/Appendix/Plonky3/.github/workflows/ci.yml b/Appendix/Plonky3/.github/workflows/ci.yml new file mode 100644 index 0000000..c18ace1 --- /dev/null +++ b/Appendix/Plonky3/.github/workflows/ci.yml @@ -0,0 +1,127 @@ +name: CI + +on: + push: + branches: [ "main" ] + pull_request: + branches: [ "*" ] + +env: + CARGO_TERM_COLOR: always + CARGO_INCREMENTAL: 1 + +jobs: + build_and_test: + name: Build and Test + runs-on: ubuntu-latest + if: "! contains(toJSON(github.event.commits.*.message), '[skip-ci]')" + + steps: + - uses: actions/checkout@v4 + + - uses: dtolnay/rust-toolchain@stable + id: rs-stable + + - uses: actions/cache@v3 + with: + path: | + ~/.cargo/bin/ + ~/.cargo/registry/index/ + ~/.cargo/registry/cache/ + ~/.cargo/git/db/ + target/ + key: rust-${{ steps.rs-stable.outputs.rustc_hash }}-${{ hashFiles('**/Cargo.toml') }} + + - name: Build + run: cargo build --verbose --all-targets + + - name: Check with parallel + run: cargo check --verbose --all-targets --features parallel + + - name: Test + run: cargo test --verbose + + - name: Test with parallel + run: cargo test --verbose --features parallel + + lint: + name: Formatting and Clippy + runs-on: ubuntu-latest + if: "! contains(toJSON(github.event.commits.*.message), '[skip-ci]')" + + steps: + - uses: actions/checkout@v4 + + - uses: dtolnay/rust-toolchain@stable + with: + components: clippy + id: rs-stable + + - uses: dtolnay/rust-toolchain@nightly + with: + components: rustfmt + + - uses: actions/cache@v3 + with: + path: | + ~/.cargo/bin/ + ~/.cargo/registry/index/ + ~/.cargo/registry/cache/ + ~/.cargo/git/db/ + target/ + key: rust-${{ steps.rs-stable.outputs.rustc_hash }}-${{ hashFiles('**/Cargo.toml') }} + + - name: Clippy + run: cargo +stable clippy --all-targets -- -D warnings + + - name: Format + run: cargo +nightly fmt --all -- --check + + check_crates: + name: Check Crates + runs-on: ubuntu-latest + if: "! contains(toJSON(github.event.commits.*.message), '[skip-ci]')" + + steps: + - uses: actions/checkout@v4 + + - uses: dtolnay/rust-toolchain@stable + id: rs-stable + + - uses: actions/cache@v3 + with: + path: | + ~/.cargo/bin/ + ~/.cargo/registry/index/ + ~/.cargo/registry/cache/ + ~/.cargo/git/db/ + target/ + key: rust-${{ steps.rs-stable.outputs.rustc_hash }}-${{ hashFiles('**/Cargo.toml', '**/Cargo.lock') }} + + - name: Cargo Check Crates + run: | + cargo check --verbose --package p3-air + cargo check --verbose --package p3-baby-bear + cargo check --verbose --package p3-blake3 + cargo check --verbose --package p3-challenger + cargo check --verbose --package p3-commit + cargo check --verbose --package p3-dft + cargo check --verbose --package p3-field + cargo check --verbose --package p3-field-testing + cargo check --verbose --package p3-fri + cargo check --verbose --package p3-goldilocks + cargo check --verbose --package p3-interpolation + cargo check --verbose --package p3-keccak + cargo check --verbose --package p3-keccak-air + cargo check --verbose --package p3-matrix + cargo check --verbose --package p3-maybe-rayon + cargo check --verbose --package p3-mds + cargo check --verbose --package p3-merkle-tree + cargo check --verbose --package p3-mersenne-31 + cargo check --verbose --package p3-monolith + cargo check --verbose --package p3-poseidon + cargo check --verbose --package p3-poseidon2 + cargo check --verbose --package p3-rescue + cargo check --verbose --package p3-symmetric + cargo check --verbose --package p3-uni-stark + cargo check --verbose --package p3-util diff --git a/Appendix/Plonky3/.gitignore b/Appendix/Plonky3/.gitignore new file mode 100644 index 0000000..d90e843 --- /dev/null +++ b/Appendix/Plonky3/.gitignore @@ -0,0 +1,3 @@ +/target +Cargo.lock +flamegraph.svg diff --git a/Appendix/Plonky3/Cargo.toml b/Appendix/Plonky3/Cargo.toml new file mode 100644 index 0000000..991a146 --- /dev/null +++ b/Appendix/Plonky3/Cargo.toml @@ -0,0 +1,38 @@ +[workspace] +resolver = "2" +members = [ + "air", + "baby-bear", + "blake3", + "bn254-fr", + "challenger", + "circle", + "commit", + "dft", + "field", + "field-testing", + "fri", + "goldilocks", + "interpolation", + "koala-bear", + "keccak", + "keccak-air", + "matrix", + "merkle-tree", + "maybe-rayon", + "mersenne-31", + "monolith", + "monty-31", + "poseidon", + "poseidon2", + "poseidon2-air", + "rescue", + "sha256", + "symmetric", + "util", + "uni-stark", +] + +[profile.profiling] +inherits = "release" +debug = true diff --git a/Appendix/Plonky3/LICENSE-APACHE b/Appendix/Plonky3/LICENSE-APACHE new file mode 100644 index 0000000..1e5006d --- /dev/null +++ b/Appendix/Plonky3/LICENSE-APACHE @@ -0,0 +1,202 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + +TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + +1. 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + +Copyright [yyyy] [name of copyright owner] + +Licensed under the Apache License, Version 2.0 (the "License"); +you may not use this file except in compliance with the License. +You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + +Unless required by applicable law or agreed to in writing, software +distributed under the License is distributed on an "AS IS" BASIS, +WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +See the License for the specific language governing permissions and +limitations under the License. + diff --git a/Appendix/Plonky3/LICENSE-MIT b/Appendix/Plonky3/LICENSE-MIT new file mode 100644 index 0000000..ae83932 --- /dev/null +++ b/Appendix/Plonky3/LICENSE-MIT @@ -0,0 +1,21 @@ +The MIT License (MIT) + +Copyright (c) 2022 The Plonky3 Authors + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in +all copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN +THE SOFTWARE. diff --git a/Appendix/Plonky3/README.md b/Appendix/Plonky3/README.md new file mode 100644 index 0000000..a91810b --- /dev/null +++ b/Appendix/Plonky3/README.md @@ -0,0 +1,206 @@ +![Plonky3-powered-by-polygon](https://github.com/Plonky3/Plonky3/assets/86010/7ec356ad-b0f3-4c4c-aa1d-3a151c1065e7) + +Plonky3 is a toolkit for implementing polynomial IOPs (PIOPs), such as PLONK and STARKs. It aims to support several polynomial commitment schemes, such as Brakedown. + +This is the "core" repo, but the plan is to move each crate into its own repo once APIs stabilize. + + +## Status + +Fields: +- [x] Mersenne31 + - [x] "complex" extension field + - [x] ~128 bit extension field + - [x] AVX2 + - [x] AVX-512 + - [x] NEON +- [x] BabyBear + - [x] ~128 bit extension field + - [x] AVX2 + - [x] AVX-512 + - [x] NEON +- [x] Goldilocks + - [x] ~128 bit extension field + +Generalized vector commitment schemes +- [x] generalized Merkle tree + +Polynomial commitment schemes +- [x] FRI-based PCS +- [ ] tensor PCS +- [ ] univariate-to-multivariate adapter +- [ ] multivariate-to-univariate adapter + +PIOPs +- [x] univariate STARK +- [ ] multivariate STARK +- [ ] PLONK + +Codes +- [x] Brakedown +- [x] Reed-Solomon + +Interpolation +- [x] Barycentric interpolation +- [x] radix-2 DIT FFT +- [x] radix-2 Bowers FFT +- [ ] four-step FFT +- [x] Mersenne circle group FFT + +Hashes +- [x] Rescue +- [x] Poseidon +- [x] Poseidon2 +- [x] BLAKE3 + - [ ] modifications to tune BLAKE3 for hashing small leaves +- [x] Keccak-256 +- [x] Monolith + + +## Benchmark + +We sometimes use a Keccak AIR to compare Plonky3's performance to other libraries like Plonky2. Several variations are possible here, with different fields and so forth, but here is one example: +``` +RUST_LOG=info cargo run --example prove_baby_bear_keccak --release --features parallel +``` + +## CPU features + +Plonky3 contains optimizations that rely on newer CPU instructions that are not available in older processors. These instruction sets include x86's [BMI1 and 2](https://en.wikipedia.org/wiki/X86_Bit_manipulation_instruction_set), [AVX2](https://en.wikipedia.org/wiki/Advanced_Vector_Extensions#Advanced_Vector_Extensions_2), and [AVX-512](https://en.wikipedia.org/wiki/AVX-512). Rustc does not emit those instructions by default; they must be explicitly enabled through the `target-feature` compiler option (or implicitly by setting `target-cpu`). To enable all features that are supported on your machine, you can set `target-cpu` to `native`. For example, to run the tests: +``` +RUSTFLAGS="-Ctarget-cpu=native" cargo test +``` + +Support for some instructions, such as AVX-512, is still experimental. They are only available in the nightly build of Rustc and are enabled by the [`nightly-features` feature flag](#nightly-only-optimizations). To use them, you must enable the flag in Rustc (e.g. by setting `target-feature`) and you must also enable the `nightly-features` feature. + + +## Nightly-only optimizations + +Some optimizations (in particular, AVX-512-optimized math) rely on features that are currently available only in the nightly build of Rustc. To use them, you need to enable the `nightly-features` feature. For example, to run the tests: +``` +cargo test --features nightly-features +``` + + +## Known issues + +The verifier might panic upon receiving certain invalid proofs. + + +## License + +Licensed under either of + +* Apache License, Version 2.0, ([LICENSE-APACHE](LICENSE-APACHE) or http://www.apache.org/licenses/LICENSE-2.0) +* MIT license ([LICENSE-MIT](LICENSE-MIT) or http://opensource.org/licenses/MIT) + +at your option. + + +## Guidance for external contributors + +Do you feel keen and able to help with Plonky3? That's great! We +encourage external contributions! + +We want to make it easy for you to contribute, but at the same time we +must manage the burden of reviewing external contributions. We are a +small team, and the time we spend reviewing external contributions is +time we are not developing ourselves. + +We also want to help you to avoid inadvertently duplicating work that +is already underway, or building something that we will not +want to incorporate. + +First and foremost, please keep in mind that this is a highly +technical piece of software and contributing is only suitable for +experienced mathematicians, cryptographers and software engineers. + +The Polygon Zero Team reserves the right to accept or reject any +external contribution for any reason, including a simple lack of time +to maintain it (now or in the future); we may even decline to review +something that is not considered a sufficiently high priority for us. + +To avoid disappointment, please communicate your intention to +contribute openly, while respecting the limited time and availability +we have to review and provide guidance for external contributions. It +is a good idea to drop a note in our public Discord #development +channel of your intention to work on something, whether an issue, a +new feature, or a performance improvement. This is probably all that's +really required to avoid duplication of work with other contributors. + +What follows are some more specific requests for how to write PRs in a +way that will make them easy for us to review. Deviating from these +guidelines may result in your PR being rejected, ignored or forgotten. + + +### General guidance for your PR + +Obviously PRs will not be considered unless they pass our Github +CI. The Github CI is not executed for PRs from forks, but you can +simulate the Github CI by running the commands in +`.github/workflows/ci.yml`. + +Under no circumstances should a single PR mix different purposes: Your +PR is either a bug fix, a new feature, or a performance improvement, +never a combination. Nor should you include, for example, two +unrelated performance improvements in one PR. Please just submit +separate PRs. The goal is to make reviewing your PR as simple as +possible, and you should be thinking about how to compose the PR to +minimise the burden on the reviewer. + +Plonky3 uses stable Rust, so any PR that depends on unstable features +is likely to be rejected. It's possible that we may relax this policy +in the future, but we aim to minimize the use of unstable features; +please discuss with us before enabling any. + +Here are a few specific guidelines for the three main categories of +PRs that we expect: + + +#### The PR fixes a bug + +In the PR description, please clearly but briefly describe + +1. the bug (could be a reference to a GH issue; if it is from a + discussion (on Discord/email/etc. for example), please copy in the + relevant parts of the discussion); +2. what turned out to the cause the bug; and +3. how the PR fixes the bug. + +Wherever possible, PRs that fix bugs should include additional tests +that (i) trigger the original bug and (ii) pass after applying the PR. + + +#### The PR implements a new feature + +If you plan to contribute an implementation of a new feature, please +double-check with the Polygon Zero team that it is a sufficient +priority for us that it will be reviewed and integrated. + +In the PR description, please clearly but briefly describe + +1. what the feature does +2. the approach taken to implement it + +All PRs for new features must include a suitable test suite. + + +#### The PR improves performance + +Performance improvements are particularly welcome! Please note that it +can be quite difficult to establish true improvements for the +workloads we care about. To help filter out false positives, the PR +description for a performance improvement must clearly identify + +1. the target bottleneck (only one per PR to avoid confusing things!) +2. how performance is measured +3. characteristics of the machine used (CPU, OS, #threads if appropriate) +4. performance before and after the PR + + +### Licensing + +Unless you explicitly state otherwise, any contribution intentionally +submitted for inclusion in the work by you, as defined in the +Apache-2.0 license, shall be dual licensed as above, without any +additional terms or conditions. diff --git a/Appendix/Plonky3/air/Cargo.toml b/Appendix/Plonky3/air/Cargo.toml new file mode 100644 index 0000000..d4bea1d --- /dev/null +++ b/Appendix/Plonky3/air/Cargo.toml @@ -0,0 +1,12 @@ +[package] +name = "p3-air" +version = "0.1.0" +edition = "2021" +license = "MIT OR Apache-2.0" + +[dependencies] +p3-field = { path = "../field" } +p3-matrix = { path = "../matrix" } + +[dev-dependencies] +p3-matrix = { path = "../matrix" } diff --git a/Appendix/Plonky3/air/src/air.rs b/Appendix/Plonky3/air/src/air.rs new file mode 100644 index 0000000..e8d7735 --- /dev/null +++ b/Appendix/Plonky3/air/src/air.rs @@ -0,0 +1,224 @@ +use core::ops::{Add, Mul, Sub}; + +use p3_field::{AbstractExtensionField, AbstractField, ExtensionField, Field}; +use p3_matrix::dense::RowMajorMatrix; +use p3_matrix::Matrix; + +/// An AIR (algebraic intermediate representation). +pub trait BaseAir: Sync { + /// The number of columns (a.k.a. registers) in this AIR. + fn width(&self) -> usize; + + fn preprocessed_trace(&self) -> Option> { + None + } +} + +/// An AIR that works with a particular `AirBuilder`. +pub trait Air: BaseAir { + fn eval(&self, builder: &mut AB); +} + +pub trait AirBuilder: Sized { + type F: Field; + + type Expr: AbstractField + + From + + Add + + Add + + Sub + + Sub + + Mul + + Mul; + + type Var: Into + + Copy + + Send + + Sync + + Add + + Add + + Add + + Sub + + Sub + + Sub + + Mul + + Mul + + Mul; + + type M: Matrix; + + fn main(&self) -> Self::M; + + fn is_first_row(&self) -> Self::Expr; + fn is_last_row(&self) -> Self::Expr; + fn is_transition(&self) -> Self::Expr { + self.is_transition_window(2) + } + fn is_transition_window(&self, size: usize) -> Self::Expr; + + /// Returns a sub-builder whose constraints are enforced only when `condition` is nonzero. + fn when>(&mut self, condition: I) -> FilteredAirBuilder<'_, Self> { + FilteredAirBuilder { + inner: self, + condition: condition.into(), + } + } + + /// Returns a sub-builder whose constraints are enforced only when `x != y`. + fn when_ne, I2: Into>( + &mut self, + x: I1, + y: I2, + ) -> FilteredAirBuilder<'_, Self> { + self.when(x.into() - y.into()) + } + + /// Returns a sub-builder whose constraints are enforced only on the first row. + fn when_first_row(&mut self) -> FilteredAirBuilder<'_, Self> { + self.when(self.is_first_row()) + } + + /// Returns a sub-builder whose constraints are enforced only on the last row. + fn when_last_row(&mut self) -> FilteredAirBuilder<'_, Self> { + self.when(self.is_last_row()) + } + + /// Returns a sub-builder whose constraints are enforced on all rows except the last. + fn when_transition(&mut self) -> FilteredAirBuilder<'_, Self> { + self.when(self.is_transition()) + } + + /// Returns a sub-builder whose constraints are enforced on all rows except the last `size - 1`. + fn when_transition_window(&mut self, size: usize) -> FilteredAirBuilder<'_, Self> { + self.when(self.is_transition_window(size)) + } + + fn assert_zero>(&mut self, x: I); + + fn assert_one>(&mut self, x: I) { + self.assert_zero(x.into() - Self::Expr::one()); + } + + fn assert_eq, I2: Into>(&mut self, x: I1, y: I2) { + self.assert_zero(x.into() - y.into()); + } + + /// Assert that `x` is a boolean, i.e. either 0 or 1. + fn assert_bool>(&mut self, x: I) { + let x = x.into(); + self.assert_zero(x.clone() * (x - Self::Expr::one())); + } +} + +pub trait AirBuilderWithPublicValues: AirBuilder { + type PublicVar: Into + Copy; + + fn public_values(&self) -> &[Self::PublicVar]; +} + +pub trait PairBuilder: AirBuilder { + fn preprocessed(&self) -> Self::M; +} + +pub trait ExtensionBuilder: AirBuilder { + type EF: ExtensionField; + + type ExprEF: AbstractExtensionField; + + type VarEF: Into + Copy + Send + Sync; + + fn assert_zero_ext(&mut self, x: I) + where + I: Into; + + fn assert_eq_ext(&mut self, x: I1, y: I2) + where + I1: Into, + I2: Into, + { + self.assert_zero_ext(x.into() - y.into()); + } + + fn assert_one_ext(&mut self, x: I) + where + I: Into, + { + self.assert_eq_ext(x, Self::ExprEF::one()) + } +} + +pub trait PermutationAirBuilder: ExtensionBuilder { + type MP: Matrix; + + type RandomVar: Into + Copy; + + fn permutation(&self) -> Self::MP; + + fn permutation_randomness(&self) -> &[Self::RandomVar]; +} + +#[derive(Debug)] +pub struct FilteredAirBuilder<'a, AB: AirBuilder> { + pub inner: &'a mut AB, + condition: AB::Expr, +} + +impl<'a, AB: AirBuilder> FilteredAirBuilder<'a, AB> { + pub fn condition(&self) -> AB::Expr { + self.condition.clone() + } +} + +impl<'a, AB: AirBuilder> AirBuilder for FilteredAirBuilder<'a, AB> { + type F = AB::F; + type Expr = AB::Expr; + type Var = AB::Var; + type M = AB::M; + + fn main(&self) -> Self::M { + self.inner.main() + } + + fn is_first_row(&self) -> Self::Expr { + self.inner.is_first_row() + } + + fn is_last_row(&self) -> Self::Expr { + self.inner.is_last_row() + } + + fn is_transition_window(&self, size: usize) -> Self::Expr { + self.inner.is_transition_window(size) + } + + fn assert_zero>(&mut self, x: I) { + self.inner.assert_zero(self.condition() * x.into()); + } +} + +impl<'a, AB: ExtensionBuilder> ExtensionBuilder for FilteredAirBuilder<'a, AB> { + type EF = AB::EF; + type ExprEF = AB::ExprEF; + type VarEF = AB::VarEF; + + fn assert_zero_ext(&mut self, x: I) + where + I: Into, + { + self.inner.assert_zero_ext(x.into() * self.condition()); + } +} + +impl<'a, AB: PermutationAirBuilder> PermutationAirBuilder for FilteredAirBuilder<'a, AB> { + type MP = AB::MP; + + type RandomVar = AB::RandomVar; + + fn permutation(&self) -> Self::MP { + self.inner.permutation() + } + + fn permutation_randomness(&self) -> &[Self::RandomVar] { + self.inner.permutation_randomness() + } +} diff --git a/Appendix/Plonky3/air/src/lib.rs b/Appendix/Plonky3/air/src/lib.rs new file mode 100644 index 0000000..e670f61 --- /dev/null +++ b/Appendix/Plonky3/air/src/lib.rs @@ -0,0 +1,11 @@ +//! APIs for AIRs, and generalizations like PAIRs. + +#![no_std] + +extern crate alloc; + +mod air; +mod virtual_column; + +pub use air::*; +pub use virtual_column::*; diff --git a/Appendix/Plonky3/air/src/virtual_column.rs b/Appendix/Plonky3/air/src/virtual_column.rs new file mode 100644 index 0000000..a9f5198 --- /dev/null +++ b/Appendix/Plonky3/air/src/virtual_column.rs @@ -0,0 +1,125 @@ +use alloc::vec; +use alloc::vec::Vec; +use core::ops::Mul; + +use p3_field::{AbstractField, Field}; + +/// An affine function over columns in a PAIR. +#[derive(Clone, Debug)] +pub struct VirtualPairCol { + column_weights: Vec<(PairCol, F)>, + constant: F, +} + +/// A column in a PAIR, i.e. either a preprocessed column or a main trace column. +#[derive(Clone, Copy, Debug)] +pub enum PairCol { + Preprocessed(usize), + Main(usize), +} + +impl PairCol { + pub const fn get(&self, preprocessed: &[T], main: &[T]) -> T { + match self { + PairCol::Preprocessed(i) => preprocessed[*i], + PairCol::Main(i) => main[*i], + } + } +} + +impl VirtualPairCol { + pub const fn new(column_weights: Vec<(PairCol, F)>, constant: F) -> Self { + Self { + column_weights, + constant, + } + } + + pub fn new_preprocessed(column_weights: Vec<(usize, F)>, constant: F) -> Self { + Self::new( + column_weights + .into_iter() + .map(|(i, w)| (PairCol::Preprocessed(i), w)) + .collect(), + constant, + ) + } + + pub fn new_main(column_weights: Vec<(usize, F)>, constant: F) -> Self { + Self::new( + column_weights + .into_iter() + .map(|(i, w)| (PairCol::Main(i), w)) + .collect(), + constant, + ) + } + + #[must_use] + pub fn one() -> Self { + Self::constant(F::one()) + } + + #[must_use] + pub const fn constant(x: F) -> Self { + Self { + column_weights: vec![], + constant: x, + } + } + + #[must_use] + pub fn single(column: PairCol) -> Self { + Self { + column_weights: vec![(column, F::one())], + constant: F::zero(), + } + } + + #[must_use] + pub fn single_preprocessed(column: usize) -> Self { + Self::single(PairCol::Preprocessed(column)) + } + + #[must_use] + pub fn single_main(column: usize) -> Self { + Self::single(PairCol::Main(column)) + } + + #[must_use] + pub fn sum_main(columns: Vec) -> Self { + let column_weights = columns.into_iter().map(|col| (col, F::one())).collect(); + Self::new_main(column_weights, F::zero()) + } + + #[must_use] + pub fn sum_preprocessed(columns: Vec) -> Self { + let column_weights = columns.into_iter().map(|col| (col, F::one())).collect(); + Self::new_preprocessed(column_weights, F::zero()) + } + + /// `a - b`, where `a` and `b` are columns in the preprocessed trace. + #[must_use] + pub fn diff_preprocessed(a_col: usize, b_col: usize) -> Self { + Self::new_preprocessed(vec![(a_col, F::one()), (b_col, F::neg_one())], F::zero()) + } + + /// `a - b`, where `a` and `b` are columns in the main trace. + #[must_use] + pub fn diff_main(a_col: usize, b_col: usize) -> Self { + Self::new_main(vec![(a_col, F::one()), (b_col, F::neg_one())], F::zero()) + } + + pub fn apply(&self, preprocessed: &[Var], main: &[Var]) -> Expr + where + F: Into, + Expr: AbstractField + Mul, + Var: Into + Copy, + { + let mut result = self.constant.into(); + for (column, weight) in self.column_weights.iter() { + result += column.get(preprocessed, main).into() * *weight; + } + result + } +} diff --git a/Appendix/Plonky3/audits/Least Authority- Polygon Plonky3 Updated Final Audit Report.pdf b/Appendix/Plonky3/audits/Least Authority- Polygon Plonky3 Updated Final Audit Report.pdf new file mode 100644 index 0000000000000000000000000000000000000000..abb5826aae36ea3c993d828ff7569d239317fabe GIT binary patch literal 284198 zcmeGEbyU<*_XmuAXBZj=5R{gb5Ts!UrI8Lr=@`1ZrG^khloDwH6+tDXB?py~6p)lq zN-61%d&cMYuHU`uuJx|XvR;$%=Do_k(hS;cwSQBZ~Wg&b1Pf|Gnn6YJVeM}bJrf_S9vgS;=fzoTnY}{NfBI@ z=Zv||5UVrh)AJ_sPAM~;{$H)Pv34e*v0HCO4m6WOEzY=6Wrd9XvqML0L1hAC+>F24 z4X{PBV|Xo_M_+tyZq*EISbXGZp)Rk!T`&77ZT|QOwPjFrtK!pz{piMqp|8Q4YZ|*F 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