hyperplonk/poly-iop/src/prod_check/mod.rs
poly-iop/src/prod_check/mod.rsRaw
1//! Main module for the Product Check protocol
2
3use crate::{
4 errors::PolyIOPErrors,
5 prod_check::util::{compute_product_poly, prove_zero_check},
6 PolyIOP, ZeroCheck,
7};
8use arithmetic::VPAuxInfo;
9use ark_ec::PairingEngine;
10use ark_ff::{One, PrimeField, Zero};
11use ark_poly::DenseMultilinearExtension;
12use ark_std::{end_timer, start_timer};
13use pcs::prelude::PolynomialCommitmentScheme;
14use std::{marker::PhantomData, rc::Rc};
15use transcript::IOPTranscript;
16
17mod util;
18
19/// A product-check proves that two n-variate multilinear polynomials `f(x),
20/// g(x)` satisfy:
21/// \prod_{x \in {0,1}^n} f(x) = \prod_{x \in {0,1}^n} g(x)
22///
23/// A ProductCheck is derived from ZeroCheck.
24///
25/// Prover steps:
26/// 1. build `prod(x0, ..., x_n)` from f and g,
27/// such that `prod(0, x1, ..., xn)` equals `f/g` over domain {0,1}^n
28/// 2. push commitments of `prod(x)` to the transcript,
29/// and `generate_challenge` from current transcript (generate alpha)
30/// 3. generate the zerocheck proof for the virtual polynomial
31/// prod(1, x) - prod(x, 0) * prod(x, 1) + alpha * (f(x) - prod(0, x) * g(x))
32///
33/// Verifier steps:
34/// 1. Extract commitments of `prod(x)` from the proof, push
35/// them to the transcript
36/// 2. `generate_challenge` from current transcript (generate alpha)
37/// 3. `verify` to verify the zerocheck proof and generate the subclaim for
38/// polynomial evaluations
39pub trait ProductCheck<E, PCS>: ZeroCheck<E::Fr>
40where
41 E: PairingEngine,
42 PCS: PolynomialCommitmentScheme<E>,
43{
44 type ProductCheckSubClaim;
45 type ProductProof;
46 type Polynomial;
47
48 /// Initialize the system with a transcript
49 ///
50 /// This function is optional -- in the case where a ProductCheck is
51 /// an building block for a more complex protocol, the transcript
52 /// may be initialized by this complex protocol, and passed to the
53 /// ProductCheck prover/verifier.
54 fn init_transcript() -> Self::Transcript;
55
56 /// Generate a proof for product check, showing that witness multilinear
57 /// polynomials f(x), g(x) satisfy `\prod_{x \in {0,1}^n} f(x) =
58 /// \prod_{x \in {0,1}^n} g(x)`
59 ///
60 /// Inputs:
61 /// - fx: the numerator multilinear polynomial
62 /// - gx: the denominator multilinear polynomial
63 /// - transcript: the IOP transcript
64 /// - pk: PCS committing key
65 ///
66 /// Outputs
67 /// - the product check proof
68 /// - the product polynomial (used for testing)
69 ///
70 /// Cost: O(N)
71 fn prove(
72 fx: &Self::Polynomial,
73 gx: &Self::Polynomial,
74 transcript: &mut IOPTranscript<E::Fr>,
75 pk: &PCS::ProverParam,
76 ) -> Result<(Self::ProductProof, Self::Polynomial), PolyIOPErrors>;
77
78 /// Verify that for witness multilinear polynomials f(x), g(x)
79 /// it holds that `\prod_{x \in {0,1}^n} f(x) = \prod_{x \in {0,1}^n} g(x)`
80 fn verify(
81 proof: &Self::ProductProof,
82 num_vars: usize,
83 transcript: &mut Self::Transcript,
84 ) -> Result<Self::ProductCheckSubClaim, PolyIOPErrors>;
85}
86
87/// A product check subclaim consists of
88/// - A zero check IOP subclaim for
89/// `Q(x) = prod(1, x) - prod(x, 0) * prod(x, 1) + challenge * (f(x) - prod(0,
90/// x) * g(x))` is 0, consists of the following:
91/// - the SubClaim from the SumCheck
92/// - the initial challenge r which is used to build eq(x, r) in ZeroCheck
93/// - The challenge `challenge`
94/// - A final query for `prod(1, ..., 1, 0) = claimed_product`.
95// Note that this final query is in fact a constant that
96// is independent from the proof. So we should avoid
97// (de)serialize it.
98#[derive(Clone, Debug, Default, PartialEq)]
99pub struct ProductCheckSubClaim<F: PrimeField, ZC: ZeroCheck<F>> {
100 // the SubClaim from the ZeroCheck
101 zero_check_sub_claim: ZC::ZeroCheckSubClaim,
102 // final query which consists of
103 // - the vector `(1, ..., 1, 0)` (needs to be reversed because Arkwork's MLE uses big-endian
104 // format for points)
105 // The expected final query evaluation is 1
106 final_query: (Vec<F>, F),
107 challenge: F,
108}
109
110/// A product check proof consists of
111/// - a zerocheck proof
112/// - a product polynomial commitment
113#[derive(Clone, Debug, Default, PartialEq)]
114pub struct ProductProof<E: PairingEngine, PCS: PolynomialCommitmentScheme<E>, ZC: ZeroCheck<E::Fr>>
115{
116 zero_check_proof: ZC::ZeroCheckProof,
117 prod_x_comm: PCS::Commitment,
118}
119
120impl<E, PCS> ProductCheck<E, PCS> for PolyIOP<E::Fr>
121where
122 E: PairingEngine,
123 PCS: PolynomialCommitmentScheme<E, Polynomial = Rc<DenseMultilinearExtension<E::Fr>>>,
124{
125 type ProductCheckSubClaim = ProductCheckSubClaim<E::Fr, Self>;
126 type ProductProof = ProductProof<E, PCS, Self>;
127 type Polynomial = Rc<DenseMultilinearExtension<E::Fr>>;
128
129 fn init_transcript() -> Self::Transcript {
130 IOPTranscript::<E::Fr>::new(b"Initializing ProductCheck transcript")
131 }
132
133 fn prove(
134 fx: &Self::Polynomial,
135 gx: &Self::Polynomial,
136 transcript: &mut IOPTranscript<E::Fr>,
137 pk: &PCS::ProverParam,
138 ) -> Result<(Self::ProductProof, Self::Polynomial), PolyIOPErrors> {
139 let start = start_timer!(|| "prod_check prove");
140
141 if fx.num_vars != gx.num_vars {
142 return Err(PolyIOPErrors::InvalidParameters(
143 "fx and gx have different number of variables".to_string(),
144 ));
145 }
146
147 // compute the product polynomial
148 let prod_x = compute_product_poly(fx, gx)?;
149
150 // generate challenge
151 let prod_x_comm = PCS::commit(pk, &Rc::new(prod_x.clone()))?;
152 transcript.append_serializable_element(b"prod(x)", &prod_x_comm)?;
153 let alpha = transcript.get_and_append_challenge(b"alpha")?;
154
155 // build the zero-check proof
156 let (zero_check_proof, _) = prove_zero_check(fx, gx, &prod_x, &alpha, transcript)?;
157
158 end_timer!(start);
159
160 Ok((
161 ProductProof {
162 zero_check_proof,
163 prod_x_comm,
164 },
165 Rc::new(prod_x.clone()),
166 ))
167 }
168
169 fn verify(
170 proof: &Self::ProductProof,
171 num_vars: usize,
172 transcript: &mut Self::Transcript,
173 ) -> Result<Self::ProductCheckSubClaim, PolyIOPErrors> {
174 let start = start_timer!(|| "prod_check verify");
175
176 // update transcript and generate challenge
177 transcript.append_serializable_element(b"prod(x)", &proof.prod_x_comm)?;
178 let alpha = transcript.get_and_append_challenge(b"alpha")?;
179
180 // invoke the zero check on the iop_proof
181 // the virtual poly info for Q(x)
182 let aux_info = VPAuxInfo {
183 max_degree: 2,
184 num_variables: num_vars,
185 phantom: PhantomData::default(),
186 };
187 let zero_check_sub_claim =
188 <Self as ZeroCheck<E::Fr>>::verify(&proof.zero_check_proof, &aux_info, transcript)?;
189
190 // the final query is on prod_x, hence has length `num_vars` + 1
191 let mut final_query = vec![E::Fr::one(); aux_info.num_variables + 1];
192 // the point has to be reversed because Arkworks uses big-endian.
193 final_query[0] = E::Fr::zero();
194 let final_eval = E::Fr::one();
195
196 end_timer!(start);
197
198 Ok(ProductCheckSubClaim {
199 zero_check_sub_claim,
200 final_query: (final_query, final_eval),
201 challenge: alpha,
202 })
203 }
204}
205
206#[cfg(test)]
207mod test {
208 use super::ProductCheck;
209 use crate::{errors::PolyIOPErrors, PolyIOP};
210 use ark_bls12_381::{Bls12_381, Fr};
211 use ark_ec::PairingEngine;
212 use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
213 use ark_std::test_rng;
214 use pcs::{prelude::KZGMultilinearPCS, PolynomialCommitmentScheme};
215 use std::rc::Rc;
216
217 // f and g are guaranteed to have the same product
218 fn test_product_check_helper<E, PCS>(
219 f: &DenseMultilinearExtension<E::Fr>,
220 g: &DenseMultilinearExtension<E::Fr>,
221 pk: &PCS::ProverParam,
222 ) -> Result<(), PolyIOPErrors>
223 where
224 E: PairingEngine,
225 PCS: PolynomialCommitmentScheme<E, Polynomial = Rc<DenseMultilinearExtension<E::Fr>>>,
226 {
227 let mut transcript = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::init_transcript();
228 transcript.append_message(b"testing", b"initializing transcript for testing")?;
229
230 let (proof, prod_x) = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::prove(
231 &Rc::new(f.clone()),
232 &Rc::new(g.clone()),
233 &mut transcript,
234 pk,
235 )?;
236
237 let mut transcript = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::init_transcript();
238 transcript.append_message(b"testing", b"initializing transcript for testing")?;
239
240 let subclaim =
241 <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::verify(&proof, f.num_vars, &mut transcript)?;
242 assert_eq!(
243 prod_x.evaluate(&subclaim.final_query.0).unwrap(),
244 subclaim.final_query.1,
245 "different product"
246 );
247
248 // bad path
249 let mut transcript = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::init_transcript();
250 transcript.append_message(b"testing", b"initializing transcript for testing")?;
251
252 let h = f + g;
253 let (bad_proof, prod_x_bad) = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::prove(
254 &Rc::new(f.clone()),
255 &Rc::new(h.clone()),
256 &mut transcript,
257 pk,
258 )?;
259
260 let mut transcript = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::init_transcript();
261 transcript.append_message(b"testing", b"initializing transcript for testing")?;
262 let bad_subclaim = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::verify(
263 &bad_proof,
264 f.num_vars,
265 &mut transcript,
266 )?;
267 assert_ne!(
268 prod_x_bad.evaluate(&bad_subclaim.final_query.0).unwrap(),
269 bad_subclaim.final_query.1,
270 "can't detect wrong proof"
271 );
272
273 Ok(())
274 }
275
276 fn test_product_check(nv: usize) -> Result<(), PolyIOPErrors> {
277 let mut rng = test_rng();
278
279 let f: DenseMultilinearExtension<Fr> = DenseMultilinearExtension::rand(nv, &mut rng);
280 let mut g = f.clone();
281 g.evaluations.reverse();
282
283 let srs = KZGMultilinearPCS::<Bls12_381>::gen_srs_for_testing(&mut rng, nv + 1)?;
284 let (pk, _) = KZGMultilinearPCS::<Bls12_381>::trim(&srs, nv + 1, Some(nv + 1))?;
285
286 test_product_check_helper::<Bls12_381, KZGMultilinearPCS<Bls12_381>>(&f, &g, &pk)?;
287
288 Ok(())
289 }
290
291 #[test]
292 fn test_trivial_polynomial() -> Result<(), PolyIOPErrors> {
293 test_product_check(1)
294 }
295 #[test]
296 fn test_normal_polynomial() -> Result<(), PolyIOPErrors> {
297 test_product_check(10)
298 }
299}