Nova/src/spartan/snark.rs
src/spartan/snark.rsRaw
1//! This module implements RelaxedR1CSSNARKTrait using Spartan that is generic
2//! over the polynomial commitment and evaluation argument (i.e., a PCS)
3//! This version of Spartan does not use preprocessing so the verifier keeps the entire
4//! description of R1CS matrices. This is essentially optimal for the verifier when using
5//! an IPA-based polynomial commitment scheme.
6
7use crate::{
8 compute_digest,
9 errors::NovaError,
10 r1cs::{R1CSShape, RelaxedR1CSInstance, RelaxedR1CSWitness},
11 spartan::{
12 polynomial::{EqPolynomial, MultilinearPolynomial, SparsePolynomial},
13 powers,
14 sumcheck::SumcheckProof,
15 PolyEvalInstance, PolyEvalWitness,
16 },
17 traits::{
18 evaluation::EvaluationEngineTrait, snark::RelaxedR1CSSNARKTrait, Group, TranscriptEngineTrait,
19 },
20 Commitment, CommitmentKey,
21};
22use ff::Field;
23use itertools::concat;
24use rayon::prelude::*;
25use serde::{Deserialize, Serialize};
26
27/// A type that represents the prover's key
28#[derive(Serialize, Deserialize)]
29#[serde(bound = "")]
30pub struct ProverKey<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
31 pk_ee: EE::ProverKey,
32 S: R1CSShape<G>,
33 vk_digest: G::Scalar, // digest of the verifier's key
34}
35
36/// A type that represents the verifier's key
37#[derive(Serialize, Deserialize)]
38#[serde(bound = "")]
39pub struct VerifierKey<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
40 vk_ee: EE::VerifierKey,
41 S: R1CSShape<G>,
42 digest: G::Scalar,
43}
44
45/// A succinct proof of knowledge of a witness to a relaxed R1CS instance
46/// The proof is produced using Spartan's combination of the sum-check and
47/// the commitment to a vector viewed as a polynomial commitment
48#[derive(Serialize, Deserialize)]
49#[serde(bound = "")]
50pub struct RelaxedR1CSSNARK<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
51 sc_proof_outer: SumcheckProof<G>,
52 claims_outer: (G::Scalar, G::Scalar, G::Scalar),
53 eval_E: G::Scalar,
54 sc_proof_inner: SumcheckProof<G>,
55 eval_W: G::Scalar,
56 sc_proof_batch: SumcheckProof<G>,
57 evals_batch: Vec<G::Scalar>,
58 eval_arg: EE::EvaluationArgument,
59}
60
61impl<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> RelaxedR1CSSNARKTrait<G>
62 for RelaxedR1CSSNARK<G, EE>
63{
64 type ProverKey = ProverKey<G, EE>;
65 type VerifierKey = VerifierKey<G, EE>;
66
67 fn setup(
68 ck: &CommitmentKey<G>,
69 S: &R1CSShape<G>,
70 ) -> Result<(Self::ProverKey, Self::VerifierKey), NovaError> {
71 let (pk_ee, vk_ee) = EE::setup(ck);
72
73 let S = S.pad();
74
75 let vk = {
76 let mut vk = VerifierKey {
77 vk_ee,
78 S: S.clone(),
79 digest: G::Scalar::ZERO,
80 };
81 vk.digest = compute_digest::<G, VerifierKey<G, EE>>(&vk);
82 vk
83 };
84
85 let pk = ProverKey {
86 pk_ee,
87 S,
88 vk_digest: vk.digest,
89 };
90
91 Ok((pk, vk))
92 }
93
94 /// produces a succinct proof of satisfiability of a RelaxedR1CS instance
95 fn prove(
96 ck: &CommitmentKey<G>,
97 pk: &Self::ProverKey,
98 U: &RelaxedR1CSInstance<G>,
99 W: &RelaxedR1CSWitness<G>,
100 ) -> Result<Self, NovaError> {
101 let W = W.pad(&pk.S); // pad the witness
102 let mut transcript = G::TE::new(b"RelaxedR1CSSNARK");
103
104 // sanity check that R1CSShape has certain size characteristics
105 pk.S.check_regular_shape();
106
107 // append the digest of vk (which includes R1CS matrices) and the RelaxedR1CSInstance to the transcript
108 transcript.absorb(b"vk", &pk.vk_digest);
109 transcript.absorb(b"U", U);
110
111 // compute the full satisfying assignment by concatenating W.W, U.u, and U.X
112 let mut z = concat(vec![W.W.clone(), vec![U.u], U.X.clone()]);
113
114 let (num_rounds_x, num_rounds_y) = (
115 (pk.S.num_cons as f64).log2() as usize,
116 ((pk.S.num_vars as f64).log2() as usize + 1),
117 );
118
119 // outer sum-check
120 let tau = (0..num_rounds_x)
121 .map(|_i| transcript.squeeze(b"t"))
122 .collect::<Result<Vec<G::Scalar>, NovaError>>()?;
123
124 let mut poly_tau = MultilinearPolynomial::new(EqPolynomial::new(tau).evals());
125 let (mut poly_Az, mut poly_Bz, poly_Cz, mut poly_uCz_E) = {
126 let (poly_Az, poly_Bz, poly_Cz) = pk.S.multiply_vec(&z)?;
127 let poly_uCz_E = (0..pk.S.num_cons)
128 .map(|i| U.u * poly_Cz[i] + W.E[i])
129 .collect::<Vec<G::Scalar>>();
130 (
131 MultilinearPolynomial::new(poly_Az),
132 MultilinearPolynomial::new(poly_Bz),
133 MultilinearPolynomial::new(poly_Cz),
134 MultilinearPolynomial::new(poly_uCz_E),
135 )
136 };
137
138 let comb_func_outer =
139 |poly_A_comp: &G::Scalar,
140 poly_B_comp: &G::Scalar,
141 poly_C_comp: &G::Scalar,
142 poly_D_comp: &G::Scalar|
143 -> G::Scalar { *poly_A_comp * (*poly_B_comp * *poly_C_comp - *poly_D_comp) };
144 let (sc_proof_outer, r_x, claims_outer) = SumcheckProof::prove_cubic_with_additive_term(
145 &G::Scalar::ZERO, // claim is zero
146 num_rounds_x,
147 &mut poly_tau,
148 &mut poly_Az,
149 &mut poly_Bz,
150 &mut poly_uCz_E,
151 comb_func_outer,
152 &mut transcript,
153 )?;
154
155 // claims from the end of sum-check
156 let (claim_Az, claim_Bz): (G::Scalar, G::Scalar) = (claims_outer[1], claims_outer[2]);
157 let claim_Cz = poly_Cz.evaluate(&r_x);
158 let eval_E = MultilinearPolynomial::new(W.E.clone()).evaluate(&r_x);
159 transcript.absorb(
160 b"claims_outer",
161 &[claim_Az, claim_Bz, claim_Cz, eval_E].as_slice(),
162 );
163
164 // inner sum-check
165 let r = transcript.squeeze(b"r")?;
166 let claim_inner_joint = claim_Az + r * claim_Bz + r * r * claim_Cz;
167
168 let poly_ABC = {
169 // compute the initial evaluation table for R(\tau, x)
170 let evals_rx = EqPolynomial::new(r_x.clone()).evals();
171
172 // Bounds "row" variables of (A, B, C) matrices viewed as 2d multilinear polynomials
173 let compute_eval_table_sparse =
174 |S: &R1CSShape<G>, rx: &[G::Scalar]| -> (Vec<G::Scalar>, Vec<G::Scalar>, Vec<G::Scalar>) {
175 assert_eq!(rx.len(), S.num_cons);
176
177 let inner = |M: &Vec<(usize, usize, G::Scalar)>, M_evals: &mut Vec<G::Scalar>| {
178 for (row, col, val) in M {
179 M_evals[*col] += rx[*row] * val;
180 }
181 };
182
183 let (A_evals, (B_evals, C_evals)) = rayon::join(
184 || {
185 let mut A_evals: Vec<G::Scalar> = vec![G::Scalar::ZERO; 2 * S.num_vars];
186 inner(&S.A, &mut A_evals);
187 A_evals
188 },
189 || {
190 rayon::join(
191 || {
192 let mut B_evals: Vec<G::Scalar> = vec![G::Scalar::ZERO; 2 * S.num_vars];
193 inner(&S.B, &mut B_evals);
194 B_evals
195 },
196 || {
197 let mut C_evals: Vec<G::Scalar> = vec![G::Scalar::ZERO; 2 * S.num_vars];
198 inner(&S.C, &mut C_evals);
199 C_evals
200 },
201 )
202 },
203 );
204
205 (A_evals, B_evals, C_evals)
206 };
207
208 let (evals_A, evals_B, evals_C) = compute_eval_table_sparse(&pk.S, &evals_rx);
209
210 assert_eq!(evals_A.len(), evals_B.len());
211 assert_eq!(evals_A.len(), evals_C.len());
212 (0..evals_A.len())
213 .into_par_iter()
214 .map(|i| evals_A[i] + r * evals_B[i] + r * r * evals_C[i])
215 .collect::<Vec<G::Scalar>>()
216 };
217
218 let poly_z = {
219 z.resize(pk.S.num_vars * 2, G::Scalar::ZERO);
220 z
221 };
222
223 let comb_func = |poly_A_comp: &G::Scalar, poly_B_comp: &G::Scalar| -> G::Scalar {
224 *poly_A_comp * *poly_B_comp
225 };
226 let (sc_proof_inner, r_y, _claims_inner) = SumcheckProof::prove_quad(
227 &claim_inner_joint,
228 num_rounds_y,
229 &mut MultilinearPolynomial::new(poly_ABC),
230 &mut MultilinearPolynomial::new(poly_z),
231 comb_func,
232 &mut transcript,
233 )?;
234
235 // add additional claims about W and E polynomials to the list from CC
236 let mut w_u_vec = Vec::new();
237 let eval_W = MultilinearPolynomial::evaluate_with(&W.W, &r_y[1..]);
238 w_u_vec.push((
239 PolyEvalWitness { p: W.W.clone() },
240 PolyEvalInstance {
241 c: U.comm_W,
242 x: r_y[1..].to_vec(),
243 e: eval_W,
244 },
245 ));
246
247 w_u_vec.push((
248 PolyEvalWitness { p: W.E },
249 PolyEvalInstance {
250 c: U.comm_E,
251 x: r_x,
252 e: eval_E,
253 },
254 ));
255
256 // We will now reduce a vector of claims of evaluations at different points into claims about them at the same point.
257 // For example, eval_W =? W(r_y[1..]) and eval_E =? E(r_x) into
258 // two claims: eval_W_prime =? W(rz) and eval_E_prime =? E(rz)
259 // We can them combine the two into one: eval_W_prime + gamma * eval_E_prime =? (W + gamma*E)(rz),
260 // where gamma is a public challenge
261 // Since commitments to W and E are homomorphic, the verifier can compute a commitment
262 // to the batched polynomial.
263 assert!(w_u_vec.len() >= 2);
264
265 let (w_vec, u_vec): (Vec<PolyEvalWitness<G>>, Vec<PolyEvalInstance<G>>) =
266 w_u_vec.into_iter().unzip();
267 let w_vec_padded = PolyEvalWitness::pad(&w_vec); // pad the polynomials to be of the same size
268 let u_vec_padded = PolyEvalInstance::pad(&u_vec); // pad the evaluation points
269
270 // generate a challenge
271 let rho = transcript.squeeze(b"r")?;
272 let num_claims = w_vec_padded.len();
273 let powers_of_rho = powers::<G>(&rho, num_claims);
274 let claim_batch_joint = u_vec_padded
275 .iter()
276 .zip(powers_of_rho.iter())
277 .map(|(u, p)| u.e * p)
278 .sum();
279
280 let mut polys_left: Vec<MultilinearPolynomial<G::Scalar>> = w_vec_padded
281 .iter()
282 .map(|w| MultilinearPolynomial::new(w.p.clone()))
283 .collect();
284 let mut polys_right: Vec<MultilinearPolynomial<G::Scalar>> = u_vec_padded
285 .iter()
286 .map(|u| MultilinearPolynomial::new(EqPolynomial::new(u.x.clone()).evals()))
287 .collect();
288
289 let num_rounds_z = u_vec_padded[0].x.len();
290 let comb_func = |poly_A_comp: &G::Scalar, poly_B_comp: &G::Scalar| -> G::Scalar {
291 *poly_A_comp * *poly_B_comp
292 };
293 let (sc_proof_batch, r_z, claims_batch) = SumcheckProof::prove_quad_batch(
294 &claim_batch_joint,
295 num_rounds_z,
296 &mut polys_left,
297 &mut polys_right,
298 &powers_of_rho,
299 comb_func,
300 &mut transcript,
301 )?;
302
303 let (claims_batch_left, _): (Vec<G::Scalar>, Vec<G::Scalar>) = claims_batch;
304
305 transcript.absorb(b"l", &claims_batch_left.as_slice());
306
307 // we now combine evaluation claims at the same point rz into one
308 let gamma = transcript.squeeze(b"g")?;
309 let powers_of_gamma: Vec<G::Scalar> = powers::<G>(&gamma, num_claims);
310 let comm_joint = u_vec_padded
311 .iter()
312 .zip(powers_of_gamma.iter())
313 .map(|(u, g_i)| u.c * *g_i)
314 .fold(Commitment::<G>::default(), |acc, item| acc + item);
315 let poly_joint = PolyEvalWitness::weighted_sum(&w_vec_padded, &powers_of_gamma);
316 let eval_joint = claims_batch_left
317 .iter()
318 .zip(powers_of_gamma.iter())
319 .map(|(e, g_i)| *e * *g_i)
320 .sum();
321
322 let eval_arg = EE::prove(
323 ck,
324 &pk.pk_ee,
325 &mut transcript,
326 &comm_joint,
327 &poly_joint.p,
328 &r_z,
329 &eval_joint,
330 )?;
331
332 Ok(RelaxedR1CSSNARK {
333 sc_proof_outer,
334 claims_outer: (claim_Az, claim_Bz, claim_Cz),
335 eval_E,
336 sc_proof_inner,
337 eval_W,
338 sc_proof_batch,
339 evals_batch: claims_batch_left,
340 eval_arg,
341 })
342 }
343
344 /// verifies a proof of satisfiability of a RelaxedR1CS instance
345 fn verify(&self, vk: &Self::VerifierKey, U: &RelaxedR1CSInstance<G>) -> Result<(), NovaError> {
346 let mut transcript = G::TE::new(b"RelaxedR1CSSNARK");
347
348 // append the digest of R1CS matrices and the RelaxedR1CSInstance to the transcript
349 transcript.absorb(b"vk", &vk.digest);
350 transcript.absorb(b"U", U);
351
352 let (num_rounds_x, num_rounds_y) = (
353 (vk.S.num_cons as f64).log2() as usize,
354 ((vk.S.num_vars as f64).log2() as usize + 1),
355 );
356
357 // outer sum-check
358 let tau = (0..num_rounds_x)
359 .map(|_i| transcript.squeeze(b"t"))
360 .collect::<Result<Vec<G::Scalar>, NovaError>>()?;
361
362 let (claim_outer_final, r_x) =
363 self
364 .sc_proof_outer
365 .verify(G::Scalar::ZERO, num_rounds_x, 3, &mut transcript)?;
366
367 // verify claim_outer_final
368 let (claim_Az, claim_Bz, claim_Cz) = self.claims_outer;
369 let taus_bound_rx = EqPolynomial::new(tau).evaluate(&r_x);
370 let claim_outer_final_expected =
371 taus_bound_rx * (claim_Az * claim_Bz - U.u * claim_Cz - self.eval_E);
372 if claim_outer_final != claim_outer_final_expected {
373 return Err(NovaError::InvalidSumcheckProof);
374 }
375
376 transcript.absorb(
377 b"claims_outer",
378 &[
379 self.claims_outer.0,
380 self.claims_outer.1,
381 self.claims_outer.2,
382 self.eval_E,
383 ]
384 .as_slice(),
385 );
386
387 // inner sum-check
388 let r = transcript.squeeze(b"r")?;
389 let claim_inner_joint =
390 self.claims_outer.0 + r * self.claims_outer.1 + r * r * self.claims_outer.2;
391
392 let (claim_inner_final, r_y) =
393 self
394 .sc_proof_inner
395 .verify(claim_inner_joint, num_rounds_y, 2, &mut transcript)?;
396
397 // verify claim_inner_final
398 let eval_Z = {
399 let eval_X = {
400 // constant term
401 let mut poly_X = vec![(0, U.u)];
402 //remaining inputs
403 poly_X.extend(
404 (0..U.X.len())
405 .map(|i| (i + 1, U.X[i]))
406 .collect::<Vec<(usize, G::Scalar)>>(),
407 );
408 SparsePolynomial::new((vk.S.num_vars as f64).log2() as usize, poly_X).evaluate(&r_y[1..])
409 };
410 (G::Scalar::ONE - r_y[0]) * self.eval_W + r_y[0] * eval_X
411 };
412
413 // compute evaluations of R1CS matrices
414 let multi_evaluate = |M_vec: &[&[(usize, usize, G::Scalar)]],
415 r_x: &[G::Scalar],
416 r_y: &[G::Scalar]|
417 -> Vec<G::Scalar> {
418 let evaluate_with_table =
419 |M: &[(usize, usize, G::Scalar)], T_x: &[G::Scalar], T_y: &[G::Scalar]| -> G::Scalar {
420 (0..M.len())
421 .collect::<Vec<usize>>()
422 .par_iter()
423 .map(|&i| {
424 let (row, col, val) = M[i];
425 T_x[row] * T_y[col] * val
426 })
427 .reduce(|| G::Scalar::ZERO, |acc, x| acc + x)
428 };
429
430 let (T_x, T_y) = rayon::join(
431 || EqPolynomial::new(r_x.to_vec()).evals(),
432 || EqPolynomial::new(r_y.to_vec()).evals(),
433 );
434
435 (0..M_vec.len())
436 .collect::<Vec<usize>>()
437 .par_iter()
438 .map(|&i| evaluate_with_table(M_vec[i], &T_x, &T_y))
439 .collect()
440 };
441
442 let evals = multi_evaluate(&[&vk.S.A, &vk.S.B, &vk.S.C], &r_x, &r_y);
443
444 let claim_inner_final_expected = (evals[0] + r * evals[1] + r * r * evals[2]) * eval_Z;
445 if claim_inner_final != claim_inner_final_expected {
446 return Err(NovaError::InvalidSumcheckProof);
447 }
448
449 // add claims about W and E polynomials
450 let u_vec: Vec<PolyEvalInstance<G>> = vec![
451 PolyEvalInstance {
452 c: U.comm_W,
453 x: r_y[1..].to_vec(),
454 e: self.eval_W,
455 },
456 PolyEvalInstance {
457 c: U.comm_E,
458 x: r_x,
459 e: self.eval_E,
460 },
461 ];
462
463 let u_vec_padded = PolyEvalInstance::pad(&u_vec); // pad the evaluation points
464
465 // generate a challenge
466 let rho = transcript.squeeze(b"r")?;
467 let num_claims = u_vec.len();
468 let powers_of_rho = powers::<G>(&rho, num_claims);
469 let claim_batch_joint = u_vec
470 .iter()
471 .zip(powers_of_rho.iter())
472 .map(|(u, p)| u.e * p)
473 .sum();
474
475 let num_rounds_z = u_vec_padded[0].x.len();
476 let (claim_batch_final, r_z) =
477 self
478 .sc_proof_batch
479 .verify(claim_batch_joint, num_rounds_z, 2, &mut transcript)?;
480
481 let claim_batch_final_expected = {
482 let poly_rz = EqPolynomial::new(r_z.clone());
483 let evals = u_vec_padded
484 .iter()
485 .map(|u| poly_rz.evaluate(&u.x))
486 .collect::<Vec<G::Scalar>>();
487
488 evals
489 .iter()
490 .zip(self.evals_batch.iter())
491 .zip(powers_of_rho.iter())
492 .map(|((e_i, p_i), rho_i)| *e_i * *p_i * rho_i)
493 .sum()
494 };
495
496 if claim_batch_final != claim_batch_final_expected {
497 return Err(NovaError::InvalidSumcheckProof);
498 }
499
500 transcript.absorb(b"l", &self.evals_batch.as_slice());
501
502 // we now combine evaluation claims at the same point rz into one
503 let gamma = transcript.squeeze(b"g")?;
504 let powers_of_gamma: Vec<G::Scalar> = powers::<G>(&gamma, num_claims);
505 let comm_joint = u_vec_padded
506 .iter()
507 .zip(powers_of_gamma.iter())
508 .map(|(u, g_i)| u.c * *g_i)
509 .fold(Commitment::<G>::default(), |acc, item| acc + item);
510 let eval_joint = self
511 .evals_batch
512 .iter()
513 .zip(powers_of_gamma.iter())
514 .map(|(e, g_i)| *e * *g_i)
515 .sum();
516
517 // verify
518 EE::verify(
519 &vk.vk_ee,
520 &mut transcript,
521 &comm_joint,
522 &r_z,
523 &eval_joint,
524 &self.eval_arg,
525 )?;
526
527 Ok(())
528 }
529}