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 zero_check::ZeroCheck,
7 PolyIOP,
8};
9use arithmetic::VPAuxInfo;
10use ark_ec::PairingEngine;
11use ark_ff::{One, PrimeField, Zero};
12use ark_poly::DenseMultilinearExtension;
13use ark_std::{end_timer, start_timer};
14use jf_primitives::pcs::prelude::PolynomialCommitmentScheme;
15use std::rc::Rc;
16use transcript::IOPTranscript;
17
18mod util;
19
20/// A product-check proves that two n-variate multilinear polynomials `f(x),
21/// g(x)` satisfy:
22/// \prod_{x \in {0,1}^n} f(x) = \prod_{x \in {0,1}^n} g(x)
23///
24/// A ProductCheck is derived from ZeroCheck.
25///
26/// Prover steps:
27/// 1. build `prod(x0, ..., x_n)` from f and g,
28/// such that `prod(0, x1, ..., xn)` equals `f/g` over domain {0,1}^n
29/// 2. push commitments of `prod(x)` to the transcript,
30/// and `generate_challenge` from current transcript (generate alpha)
31/// 3. generate the zerocheck proof for the virtual polynomial
32/// prod(1, x) - prod(x, 0) * prod(x, 1) + alpha * (f(x) - prod(0, x) * g(x))
33///
34/// Verifier steps:
35/// 1. Extract commitments of `prod(x)` from the proof, push
36/// them to the transcript
37/// 2. `generate_challenge` from current transcript (generate alpha)
38/// 3. `verify` to verify the zerocheck proof and generate the subclaim for
39/// polynomial evaluations
40pub trait ProductCheck<E, PCS>: ZeroCheck<E::Fr>
41where
42 E: PairingEngine,
43 PCS: PolynomialCommitmentScheme<E>,
44{
45 type ProductCheckSubClaim;
46 type ProductCheckProof;
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 pcs_param: &PCS::ProverParam,
73 fx: &Self::MultilinearExtension,
74 gx: &Self::MultilinearExtension,
75 transcript: &mut IOPTranscript<E::Fr>,
76 ) -> Result<(Self::ProductCheckProof, Self::MultilinearExtension), 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::ProductCheckProof,
82 aux_info: &VPAuxInfo<E::Fr>,
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) + alpha * (f(x) - prod(0,
90/// x) * g(x)) = 0`
91/// - The random challenge `alpha`
92/// - A final query for `prod(1, ..., 1, 0) = 1`.
93// Note that this final query is in fact a constant that
94// is independent from the proof. So we should avoid
95// (de)serialize it.
96#[derive(Clone, Debug, Default, PartialEq)]
97pub struct ProductCheckSubClaim<F: PrimeField, ZC: ZeroCheck<F>> {
98 // the SubClaim from the ZeroCheck
99 pub zero_check_sub_claim: ZC::ZeroCheckSubClaim,
100 // final query which consists of
101 // - the vector `(1, ..., 1, 0)` (needs to be reversed because Arkwork's MLE uses big-endian
102 // format for points)
103 // The expected final query evaluation is 1
104 pub final_query: (Vec<F>, F),
105 pub alpha: F,
106}
107
108/// A product check proof consists of
109/// - a zerocheck proof
110/// - a product polynomial commitment
111#[derive(Clone, Debug, Default, PartialEq)]
112pub struct ProductCheckProof<
113 E: PairingEngine,
114 PCS: PolynomialCommitmentScheme<E>,
115 ZC: ZeroCheck<E::Fr>,
116> {
117 pub zero_check_proof: ZC::ZeroCheckProof,
118 pub prod_x_comm: PCS::Commitment,
119}
120
121impl<E, PCS> ProductCheck<E, PCS> for PolyIOP<E::Fr>
122where
123 E: PairingEngine,
124 PCS: PolynomialCommitmentScheme<E, Polynomial = Rc<DenseMultilinearExtension<E::Fr>>>,
125{
126 type ProductCheckSubClaim = ProductCheckSubClaim<E::Fr, Self>;
127 type ProductCheckProof = ProductCheckProof<E, PCS, Self>;
128
129 fn init_transcript() -> Self::Transcript {
130 IOPTranscript::<E::Fr>::new(b"Initializing ProductCheck transcript")
131 }
132
133 fn prove(
134 pcs_param: &PCS::ProverParam,
135 fx: &Self::MultilinearExtension,
136 gx: &Self::MultilinearExtension,
137 transcript: &mut IOPTranscript<E::Fr>,
138 ) -> Result<(Self::ProductCheckProof, Self::MultilinearExtension), 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(pcs_param, &prod_x)?;
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 ProductCheckProof {
162 zero_check_proof,
163 prod_x_comm,
164 },
165 prod_x,
166 ))
167 }
168
169 fn verify(
170 proof: &Self::ProductCheckProof,
171 aux_info: &VPAuxInfo<E::Fr>,
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 zero_check_sub_claim =
183 <Self as ZeroCheck<E::Fr>>::verify(&proof.zero_check_proof, aux_info, transcript)?;
184
185 // the final query is on prod_x, hence has length `num_vars` + 1
186 let mut final_query = vec![E::Fr::one(); aux_info.num_variables + 1];
187 // the point has to be reversed because Arkworks uses big-endian.
188 final_query[0] = E::Fr::zero();
189 let final_eval = E::Fr::one();
190
191 end_timer!(start);
192
193 Ok(ProductCheckSubClaim {
194 zero_check_sub_claim,
195 final_query: (final_query, final_eval),
196 alpha,
197 })
198 }
199}
200
201#[cfg(test)]
202mod test {
203 use super::ProductCheck;
204 use crate::{errors::PolyIOPErrors, PolyIOP};
205 use arithmetic::VPAuxInfo;
206 use ark_bls12_381::{Bls12_381, Fr};
207 use ark_ec::PairingEngine;
208 use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
209 use ark_std::test_rng;
210 use jf_primitives::pcs::{prelude::MultilinearKzgPCS, PolynomialCommitmentScheme};
211 use std::{marker::PhantomData, rc::Rc};
212
213 // f and g are guaranteed to have the same product
214 fn test_product_check_helper<E, PCS>(
215 f: &DenseMultilinearExtension<E::Fr>,
216 g: &DenseMultilinearExtension<E::Fr>,
217 pcs_param: &PCS::ProverParam,
218 ) -> Result<(), PolyIOPErrors>
219 where
220 E: PairingEngine,
221 PCS: PolynomialCommitmentScheme<E, Polynomial = Rc<DenseMultilinearExtension<E::Fr>>>,
222 {
223 let mut transcript = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::init_transcript();
224 transcript.append_message(b"testing", b"initializing transcript for testing")?;
225
226 let (proof, prod_x) = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::prove(
227 pcs_param,
228 &Rc::new(f.clone()),
229 &Rc::new(g.clone()),
230 &mut transcript,
231 )?;
232
233 let mut transcript = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::init_transcript();
234 transcript.append_message(b"testing", b"initializing transcript for testing")?;
235
236 let aux_info = VPAuxInfo {
237 max_degree: 2,
238 num_variables: f.num_vars,
239 phantom: PhantomData::default(),
240 };
241 let prod_subclaim =
242 <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::verify(&proof, &aux_info, &mut transcript)?;
243 assert_eq!(
244 prod_x.evaluate(&prod_subclaim.final_query.0).unwrap(),
245 prod_subclaim.final_query.1,
246 "different product"
247 );
248
249 // bad path
250 let mut transcript = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::init_transcript();
251 transcript.append_message(b"testing", b"initializing transcript for testing")?;
252
253 let h = f + g;
254 let (bad_proof, prod_x_bad) = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::prove(
255 pcs_param,
256 &Rc::new(f.clone()),
257 &Rc::new(h),
258 &mut transcript,
259 )?;
260
261 let mut transcript = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::init_transcript();
262 transcript.append_message(b"testing", b"initializing transcript for testing")?;
263 let bad_subclaim = <PolyIOP<E::Fr> as ProductCheck<E, PCS>>::verify(
264 &bad_proof,
265 &aux_info,
266 &mut transcript,
267 )?;
268 assert_ne!(
269 prod_x_bad.evaluate(&bad_subclaim.final_query.0).unwrap(),
270 bad_subclaim.final_query.1,
271 "can't detect wrong proof"
272 );
273
274 Ok(())
275 }
276
277 fn test_product_check(nv: usize) -> Result<(), PolyIOPErrors> {
278 let mut rng = test_rng();
279
280 let f: DenseMultilinearExtension<Fr> = DenseMultilinearExtension::rand(nv, &mut rng);
281 let mut g = f.clone();
282 g.evaluations.reverse();
283
284 let srs = MultilinearKzgPCS::<Bls12_381>::gen_srs_for_testing(&mut rng, nv + 1)?;
285 let (pcs_param, _) = MultilinearKzgPCS::<Bls12_381>::trim(&srs, nv + 1, Some(nv + 1))?;
286
287 test_product_check_helper::<Bls12_381, MultilinearKzgPCS<Bls12_381>>(&f, &g, &pcs_param)?;
288
289 Ok(())
290 }
291
292 #[test]
293 fn test_trivial_polynomial() -> Result<(), PolyIOPErrors> {
294 test_product_check(1)
295 }
296 #[test]
297 fn test_normal_polynomial() -> Result<(), PolyIOPErrors> {
298 test_product_check(10)
299 }
300}