testudo/src/r1csproof.rs
src/r1csproof.rsRaw
1#![allow(clippy::too_many_arguments)]
2use super::commitments::{Commitments, MultiCommitGens};
3use super::dense_mlpoly::{
4 DensePolynomial, EqPolynomial, PolyCommitment, PolyCommitmentGens, PolyEvalProof,
5};
6use super::errors::ProofVerifyError;
7use super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
8use super::nizk::{EqualityProof, KnowledgeProof, ProductProof};
9use super::r1csinstance::R1CSInstance;
10use super::random::RandomTape;
11use super::scalar::Scalar;
12use super::sparse_mlpoly::{SparsePolyEntry, SparsePolynomial};
13use super::sumcheck::ZKSumcheckInstanceProof;
14use super::timer::Timer;
15use super::transcript::{AppendToTranscript, ProofTranscript};
16use core::iter;
17use merlin::Transcript;
18use ark_serialize::*;
19use ark_std::{Zero, One};
20
21#[derive(CanonicalSerialize, CanonicalDeserialize, Debug)]
22pub struct R1CSProof {
23 comm_vars: PolyCommitment,
24 sc_proof_phase1: ZKSumcheckInstanceProof,
25 claims_phase2: (
26 CompressedGroup,
27 CompressedGroup,
28 CompressedGroup,
29 CompressedGroup,
30 ),
31 pok_claims_phase2: (KnowledgeProof, ProductProof),
32 proof_eq_sc_phase1: EqualityProof,
33 sc_proof_phase2: ZKSumcheckInstanceProof,
34 comm_vars_at_ry: CompressedGroup,
35 proof_eval_vars_at_ry: PolyEvalProof,
36 proof_eq_sc_phase2: EqualityProof,
37}
38
39pub struct R1CSSumcheckGens {
40 gens_1: MultiCommitGens,
41 gens_3: MultiCommitGens,
42 gens_4: MultiCommitGens,
43}
44
45// TODO: fix passing gens_1_ref
46impl R1CSSumcheckGens {
47 pub fn new(label: &'static [u8], gens_1_ref: &MultiCommitGens) -> Self {
48 let gens_1 = gens_1_ref.clone();
49 let gens_3 = MultiCommitGens::new(3, label);
50 let gens_4 = MultiCommitGens::new(4, label);
51
52 R1CSSumcheckGens {
53 gens_1,
54 gens_3,
55 gens_4,
56 }
57 }
58}
59
60pub struct R1CSGens {
61 gens_sc: R1CSSumcheckGens,
62 gens_pc: PolyCommitmentGens,
63}
64
65impl R1CSGens {
66 pub fn new(label: &'static [u8], _num_cons: usize, num_vars: usize) -> Self {
67 let num_poly_vars = num_vars.log2() as usize;
68 let gens_pc = PolyCommitmentGens::new(num_poly_vars, label);
69 let gens_sc = R1CSSumcheckGens::new(label, &gens_pc.gens.gens_1);
70 R1CSGens { gens_sc, gens_pc }
71 }
72}
73
74impl R1CSProof {
75 fn prove_phase_one(
76 num_rounds: usize,
77 evals_tau: &mut DensePolynomial,
78 evals_Az: &mut DensePolynomial,
79 evals_Bz: &mut DensePolynomial,
80 evals_Cz: &mut DensePolynomial,
81 gens: &R1CSSumcheckGens,
82 transcript: &mut Transcript,
83 random_tape: &mut RandomTape,
84 ) -> (ZKSumcheckInstanceProof, Vec<Scalar>, Vec<Scalar>, Scalar) {
85 let comb_func = |poly_A_comp: &Scalar,
86 poly_B_comp: &Scalar,
87 poly_C_comp: &Scalar,
88 poly_D_comp: &Scalar|
89 -> Scalar { poly_A_comp * (poly_B_comp * poly_C_comp - poly_D_comp) };
90
91 let (sc_proof_phase_one, r, claims, blind_claim_postsc) =
92 ZKSumcheckInstanceProof::prove_cubic_with_additive_term(
93 &Scalar::zero(), // claim is zero
94 &Scalar::zero(), // blind for claim is also zero
95 num_rounds,
96 evals_tau,
97 evals_Az,
98 evals_Bz,
99 evals_Cz,
100 comb_func,
101 &gens.gens_1,
102 &gens.gens_4,
103 transcript,
104 random_tape,
105 );
106
107 (sc_proof_phase_one, r, claims, blind_claim_postsc)
108 }
109
110 fn prove_phase_two(
111 num_rounds: usize,
112 claim: &Scalar,
113 blind_claim: &Scalar,
114 evals_z: &mut DensePolynomial,
115 evals_ABC: &mut DensePolynomial,
116 gens: &R1CSSumcheckGens,
117 transcript: &mut Transcript,
118 random_tape: &mut RandomTape,
119 ) -> (ZKSumcheckInstanceProof, Vec<Scalar>, Vec<Scalar>, Scalar) {
120 let comb_func =
121 |poly_A_comp: &Scalar, poly_B_comp: &Scalar| -> Scalar { poly_A_comp * poly_B_comp };
122 let (sc_proof_phase_two, r, claims, blind_claim_postsc) = ZKSumcheckInstanceProof::prove_quad(
123 claim,
124 blind_claim,
125 num_rounds,
126 evals_z,
127 evals_ABC,
128 comb_func,
129 &gens.gens_1,
130 &gens.gens_3,
131 transcript,
132 random_tape,
133 );
134
135 (sc_proof_phase_two, r, claims, blind_claim_postsc)
136 }
137
138 fn protocol_name() -> &'static [u8] {
139 b"R1CS proof"
140 }
141
142 pub fn prove(
143 inst: &R1CSInstance,
144 vars: Vec<Scalar>,
145 input: &[Scalar],
146 gens: &R1CSGens,
147 transcript: &mut Transcript,
148 random_tape: &mut RandomTape,
149 ) -> (R1CSProof, Vec<Scalar>, Vec<Scalar>) {
150 let timer_prove = Timer::new("R1CSProof::prove");
151 transcript.append_protocol_name(R1CSProof::protocol_name());
152
153 // we currently require the number of |inputs| + 1 to be at most number of vars
154 assert!(input.len() < vars.len());
155
156 input.append_to_transcript(b"input", transcript);
157
158 let timer_commit = Timer::new("polycommit");
159 let (poly_vars, comm_vars, blinds_vars) = {
160 // create a multilinear polynomial using the supplied assignment for variables
161 let poly_vars = DensePolynomial::new(vars.clone());
162
163 // produce a commitment to the satisfying assignment
164 let (comm_vars, blinds_vars) = poly_vars.commit(&gens.gens_pc, Some(random_tape));
165
166 // add the commitment to the prover's transcript
167 comm_vars.append_to_transcript(b"poly_commitment", transcript);
168 (poly_vars, comm_vars, blinds_vars)
169 };
170 timer_commit.stop();
171
172 let timer_sc_proof_phase1 = Timer::new("prove_sc_phase_one");
173
174 // append input to variables to create a single vector z
175 let z = {
176 let num_inputs = input.len();
177 let num_vars = vars.len();
178 let mut z = vars;
179 z.extend(&vec![Scalar::one()]); // add constant term in z
180 z.extend(input);
181 z.extend(&vec![Scalar::zero(); num_vars - num_inputs - 1]); // we will pad with zeros
182 z
183 };
184
185 // derive the verifier's challenge tau
186 let (num_rounds_x, num_rounds_y) =
187 (inst.get_num_cons().log2() as usize, z.len().log2() as usize);
188 let tau = transcript.challenge_vector(b"challenge_tau", num_rounds_x);
189 // compute the initial evaluation table for R(\tau, x)
190 let mut poly_tau = DensePolynomial::new(EqPolynomial::new(tau).evals());
191 let (mut poly_Az, mut poly_Bz, mut poly_Cz) =
192 inst.multiply_vec(inst.get_num_cons(), z.len(), &z);
193
194 let (sc_proof_phase1, rx, _claims_phase1, blind_claim_postsc1) = R1CSProof::prove_phase_one(
195 num_rounds_x,
196 &mut poly_tau,
197 &mut poly_Az,
198 &mut poly_Bz,
199 &mut poly_Cz,
200 &gens.gens_sc,
201 transcript,
202 random_tape,
203 );
204 assert_eq!(poly_tau.len(), 1);
205 assert_eq!(poly_Az.len(), 1);
206 assert_eq!(poly_Bz.len(), 1);
207 assert_eq!(poly_Cz.len(), 1);
208 timer_sc_proof_phase1.stop();
209
210 let (tau_claim, Az_claim, Bz_claim, Cz_claim) =
211 (&poly_tau[0], &poly_Az[0], &poly_Bz[0], &poly_Cz[0]);
212 let (Az_blind, Bz_blind, Cz_blind, prod_Az_Bz_blind) = (
213 random_tape.random_scalar(b"Az_blind"),
214 random_tape.random_scalar(b"Bz_blind"),
215 random_tape.random_scalar(b"Cz_blind"),
216 random_tape.random_scalar(b"prod_Az_Bz_blind"),
217 );
218
219 let (pok_Cz_claim, comm_Cz_claim) = {
220 KnowledgeProof::prove(
221 &gens.gens_sc.gens_1,
222 transcript,
223 random_tape,
224 Cz_claim,
225 &Cz_blind,
226 )
227 };
228
229 let (proof_prod, comm_Az_claim, comm_Bz_claim, comm_prod_Az_Bz_claims) = {
230 let prod = Az_claim * Bz_claim;
231 ProductProof::prove(
232 &gens.gens_sc.gens_1,
233 transcript,
234 random_tape,
235 Az_claim,
236 &Az_blind,
237 Bz_claim,
238 &Bz_blind,
239 &prod,
240 &prod_Az_Bz_blind,
241 )
242 };
243
244 comm_Az_claim.append_to_transcript(b"comm_Az_claim", transcript);
245 comm_Bz_claim.append_to_transcript(b"comm_Bz_claim", transcript);
246 comm_Cz_claim.append_to_transcript(b"comm_Cz_claim", transcript);
247 comm_prod_Az_Bz_claims.append_to_transcript(b"comm_prod_Az_Bz_claims", transcript);
248
249 // prove the final step of sum-check #1
250 let taus_bound_rx = tau_claim;
251 let blind_expected_claim_postsc1 = taus_bound_rx * (prod_Az_Bz_blind - Cz_blind);
252 let claim_post_phase1 = (Az_claim * Bz_claim - Cz_claim) * taus_bound_rx;
253 let (proof_eq_sc_phase1, _C1, _C2) = EqualityProof::prove(
254 &gens.gens_sc.gens_1,
255 transcript,
256 random_tape,
257 &claim_post_phase1,
258 &blind_expected_claim_postsc1,
259 &claim_post_phase1,
260 &blind_claim_postsc1,
261 );
262
263 let timer_sc_proof_phase2 = Timer::new("prove_sc_phase_two");
264 // combine the three claims into a single claim
265 let r_A = transcript.challenge_scalar(b"challenege_Az");
266 let r_B = transcript.challenge_scalar(b"challenege_Bz");
267 let r_C = transcript.challenge_scalar(b"challenege_Cz");
268 let claim_phase2 = r_A * Az_claim + r_B * Bz_claim + r_C * Cz_claim;
269 let blind_claim_phase2 = r_A * Az_blind + r_B * Bz_blind + r_C * Cz_blind;
270
271 let evals_ABC = {
272 // compute the initial evaluation table for R(\tau, x)
273 let evals_rx = EqPolynomial::new(rx.clone()).evals();
274 let (evals_A, evals_B, evals_C) =
275 inst.compute_eval_table_sparse(inst.get_num_cons(), z.len(), &evals_rx);
276
277 assert_eq!(evals_A.len(), evals_B.len());
278 assert_eq!(evals_A.len(), evals_C.len());
279 (0..evals_A.len())
280 .map(|i| r_A * evals_A[i] + r_B * evals_B[i] + r_C * evals_C[i])
281 .collect::<Vec<Scalar>>()
282 };
283
284 // another instance of the sum-check protocol
285 let (sc_proof_phase2, ry, claims_phase2, blind_claim_postsc2) = R1CSProof::prove_phase_two(
286 num_rounds_y,
287 &claim_phase2,
288 &blind_claim_phase2,
289 &mut DensePolynomial::new(z),
290 &mut DensePolynomial::new(evals_ABC),
291 &gens.gens_sc,
292 transcript,
293 random_tape,
294 );
295 timer_sc_proof_phase2.stop();
296
297 let timer_polyeval = Timer::new("polyeval");
298 let eval_vars_at_ry = poly_vars.evaluate(&ry[1..].to_vec());
299 let blind_eval = random_tape.random_scalar(b"blind_eval");
300 let (proof_eval_vars_at_ry, comm_vars_at_ry) = PolyEvalProof::prove(
301 &poly_vars,
302 Some(&blinds_vars),
303 &ry[1..].to_vec(),
304 &eval_vars_at_ry,
305 Some(&blind_eval),
306 &gens.gens_pc,
307 transcript,
308 random_tape,
309 );
310 timer_polyeval.stop();
311
312 // prove the final step of sum-check #2
313 let blind_eval_Z_at_ry = (Scalar::one() - ry[0]) * blind_eval;
314 let blind_expected_claim_postsc2 = claims_phase2[1] * blind_eval_Z_at_ry;
315 let claim_post_phase2 = claims_phase2[0] * claims_phase2[1];
316 let (proof_eq_sc_phase2, _C1, _C2) = EqualityProof::prove(
317 &gens.gens_pc.gens.gens_1,
318 transcript,
319 random_tape,
320 &claim_post_phase2,
321 &blind_expected_claim_postsc2,
322 &claim_post_phase2,
323 &blind_claim_postsc2,
324 );
325
326 timer_prove.stop();
327
328 (
329 R1CSProof {
330 comm_vars,
331 sc_proof_phase1,
332 claims_phase2: (
333 comm_Az_claim,
334 comm_Bz_claim,
335 comm_Cz_claim,
336 comm_prod_Az_Bz_claims,
337 ),
338 pok_claims_phase2: (pok_Cz_claim, proof_prod),
339 proof_eq_sc_phase1,
340 sc_proof_phase2,
341 comm_vars_at_ry,
342 proof_eval_vars_at_ry,
343 proof_eq_sc_phase2,
344 },
345 rx,
346 ry,
347 )
348 }
349
350 pub fn verify(
351 &self,
352 num_vars: usize,
353 num_cons: usize,
354 input: &[Scalar],
355 evals: &(Scalar, Scalar, Scalar),
356 transcript: &mut Transcript,
357 gens: &R1CSGens,
358 ) -> Result<(Vec<Scalar>, Vec<Scalar>), ProofVerifyError> {
359 transcript.append_protocol_name(R1CSProof::protocol_name());
360
361 input.append_to_transcript(b"input", transcript);
362
363 let n = num_vars;
364 // add the commitment to the verifier's transcript
365 self
366 .comm_vars
367 .append_to_transcript(b"poly_commitment", transcript);
368
369 let (num_rounds_x, num_rounds_y) = (num_cons.log2() as usize, (2 * num_vars).log2() as usize);
370
371 // derive the verifier's challenge tau
372 let tau = transcript.challenge_vector(b"challenge_tau", num_rounds_x);
373
374 // verify the first sum-check instance
375 let claim_phase1 = Scalar::zero()
376 .commit(&Scalar::zero(), &gens.gens_sc.gens_1)
377 .compress();
378 let (comm_claim_post_phase1, rx) = self.sc_proof_phase1.verify(
379 &claim_phase1,
380 num_rounds_x,
381 3,
382 &gens.gens_sc.gens_1,
383 &gens.gens_sc.gens_4,
384 transcript,
385 )?;
386 // perform the intermediate sum-check test with claimed Az, Bz, and Cz
387 let (comm_Az_claim, comm_Bz_claim, comm_Cz_claim, comm_prod_Az_Bz_claims) = &self.claims_phase2;
388 let (pok_Cz_claim, proof_prod) = &self.pok_claims_phase2;
389
390 pok_Cz_claim.verify(&gens.gens_sc.gens_1, transcript, comm_Cz_claim)?;
391 proof_prod.verify(
392 &gens.gens_sc.gens_1,
393 transcript,
394 comm_Az_claim,
395 comm_Bz_claim,
396 comm_prod_Az_Bz_claims,
397 )?;
398
399 comm_Az_claim.append_to_transcript(b"comm_Az_claim", transcript);
400 comm_Bz_claim.append_to_transcript(b"comm_Bz_claim", transcript);
401 comm_Cz_claim.append_to_transcript(b"comm_Cz_claim", transcript);
402 comm_prod_Az_Bz_claims.append_to_transcript(b"comm_prod_Az_Bz_claims", transcript);
403
404 let taus_bound_rx: Scalar = (0..rx.len())
405 .map(|i| rx[i] * tau[i] + (Scalar::one() - rx[i]) * (Scalar::one() - tau[i]))
406 .product();
407 let expected_claim_post_phase1 = (taus_bound_rx
408 * (comm_prod_Az_Bz_claims.decompress().unwrap() - comm_Cz_claim.decompress().unwrap()))
409 .compress();
410
411 // verify proof that expected_claim_post_phase1 == claim_post_phase1
412 self.proof_eq_sc_phase1.verify(
413 &gens.gens_sc.gens_1,
414 transcript,
415 &expected_claim_post_phase1,
416 &comm_claim_post_phase1,
417 )?;
418
419 // derive three public challenges and then derive a joint claim
420 let r_A = transcript.challenge_scalar(b"challenege_Az");
421 let r_B = transcript.challenge_scalar(b"challenege_Bz");
422 let r_C = transcript.challenge_scalar(b"challenege_Cz");
423
424 // r_A * comm_Az_claim + r_B * comm_Bz_claim + r_C * comm_Cz_claim;
425 let comm_claim_phase2 = GroupElement::vartime_multiscalar_mul(
426 iter::once(&r_A)
427 .chain(iter::once(&r_B))
428 .chain(iter::once(&r_C)),
429 iter::once(&comm_Az_claim)
430 .chain(iter::once(&comm_Bz_claim))
431 .chain(iter::once(&comm_Cz_claim))
432 .map(|pt| pt.decompress().unwrap())
433 .collect::<Vec<GroupElement>>(),
434 )
435 .compress();
436
437 // verify the joint claim with a sum-check protocol
438 let (comm_claim_post_phase2, ry) = self.sc_proof_phase2.verify(
439 &comm_claim_phase2,
440 num_rounds_y,
441 2,
442 &gens.gens_sc.gens_1,
443 &gens.gens_sc.gens_3,
444 transcript,
445 )?;
446
447 // verify Z(ry) proof against the initial commitment
448 self.proof_eval_vars_at_ry.verify(
449 &gens.gens_pc,
450 transcript,
451 &ry[1..].to_vec(),
452 &self.comm_vars_at_ry,
453 &self.comm_vars,
454 )?;
455
456 let poly_input_eval = {
457 // constant term
458 let mut input_as_sparse_poly_entries = vec![SparsePolyEntry::new(0, Scalar::one())];
459 //remaining inputs
460 input_as_sparse_poly_entries.extend(
461 (0..input.len())
462 .map(|i| SparsePolyEntry::new(i + 1, input[i]))
463 .collect::<Vec<SparsePolyEntry>>(),
464 );
465 SparsePolynomial::new(n.log2() as usize, input_as_sparse_poly_entries)
466 .evaluate(&ry[1..].to_vec())
467 };
468
469 // compute commitment to eval_Z_at_ry = (Scalar::one() - ry[0]) * self.eval_vars_at_ry + ry[0] * poly_input_eval
470 let comm_eval_Z_at_ry = GroupElement::vartime_multiscalar_mul(
471 iter::once(Scalar::one() - ry[0]).chain(iter::once(ry[0])),
472 iter::once(&self.comm_vars_at_ry.decompress().unwrap()).chain(iter::once(
473 &poly_input_eval.commit(&Scalar::zero(), &gens.gens_pc.gens.gens_1),
474 )),
475 );
476
477 // perform the final check in the second sum-check protocol
478 let (eval_A_r, eval_B_r, eval_C_r) = evals;
479 let expected_claim_post_phase2 =
480 ((r_A * eval_A_r + r_B * eval_B_r + r_C * eval_C_r) * comm_eval_Z_at_ry).compress();
481 // verify proof that expected_claim_post_phase1 == claim_post_phase1
482 self.proof_eq_sc_phase2.verify(
483 &gens.gens_sc.gens_1,
484 transcript,
485 &expected_claim_post_phase2,
486 &comm_claim_post_phase2,
487 )?;
488
489 Ok((rx, ry))
490 }
491}
492
493#[cfg(test)]
494mod tests {
495 use super::*;
496use ark_std::{UniformRand};
497
498 fn produce_tiny_r1cs() -> (R1CSInstance, Vec<Scalar>, Vec<Scalar>) {
499 // three constraints over five variables Z1, Z2, Z3, Z4, and Z5
500 // rounded to the nearest power of two
501 let num_cons = 128;
502 let num_vars = 256;
503 let num_inputs = 2;
504
505 // encode the above constraints into three matrices
506 let mut A: Vec<(usize, usize, Scalar)> = Vec::new();
507 let mut B: Vec<(usize, usize, Scalar)> = Vec::new();
508 let mut C: Vec<(usize, usize, Scalar)> = Vec::new();
509
510 let one = Scalar::one();
511 // constraint 0 entries
512 // (Z1 + Z2) * I0 - Z3 = 0;
513 A.push((0, 0, one));
514 A.push((0, 1, one));
515 B.push((0, num_vars + 1, one));
516 C.push((0, 2, one));
517
518 // constraint 1 entries
519 // (Z1 + I1) * (Z3) - Z4 = 0
520 A.push((1, 0, one));
521 A.push((1, num_vars + 2, one));
522 B.push((1, 2, one));
523 C.push((1, 3, one));
524 // constraint 3 entries
525 // Z5 * 1 - 0 = 0
526 A.push((2, 4, one));
527 B.push((2, num_vars, one));
528
529 let inst = R1CSInstance::new(num_cons, num_vars, num_inputs, &A, &B, &C);
530
531 // compute a satisfying assignment
532 let mut rng = ark_std::rand::thread_rng();
533 let i0 = Scalar::rand(&mut rng);
534 let i1 = Scalar::rand(&mut rng);
535 let z1 = Scalar::rand(&mut rng);
536 let z2 = Scalar::rand(&mut rng);
537 let z3 = (z1 + z2) * i0; // constraint 1: (Z1 + Z2) * I0 - Z3 = 0;
538 let z4 = (z1 + i1) * z3; // constraint 2: (Z1 + I1) * (Z3) - Z4 = 0
539 let z5 = Scalar::zero(); //constraint 3
540
541 let mut vars = vec![Scalar::zero(); num_vars];
542 vars[0] = z1;
543 vars[1] = z2;
544 vars[2] = z3;
545 vars[3] = z4;
546 vars[4] = z5;
547
548 let mut input = vec![Scalar::zero(); num_inputs];
549 input[0] = i0;
550 input[1] = i1;
551
552 (inst, vars, input)
553 }
554
555 #[test]
556 fn test_tiny_r1cs() {
557 let (inst, vars, input) = tests::produce_tiny_r1cs();
558 let is_sat = inst.is_sat(&vars, &input);
559 assert!(is_sat);
560 }
561
562 #[test]
563 fn test_synthetic_r1cs() {
564 let (inst, vars, input) = R1CSInstance::produce_synthetic_r1cs(1024, 1024, 10);
565 let is_sat = inst.is_sat(&vars, &input);
566 assert!(is_sat);
567 }
568
569 #[test]
570 pub fn check_r1cs_proof() {
571 let num_vars = 1024;
572 let num_cons = num_vars;
573 let num_inputs = 10;
574 let (inst, vars, input) = R1CSInstance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
575
576 let gens = R1CSGens::new(b"test-m", num_cons, num_vars);
577
578 let mut random_tape = RandomTape::new(b"proof");
579 let mut prover_transcript = Transcript::new(b"example");
580 let (proof, rx, ry) = R1CSProof::prove(
581 &inst,
582 vars,
583 &input,
584 &gens,
585 &mut prover_transcript,
586 &mut random_tape,
587 );
588
589 let inst_evals = inst.evaluate(&rx, &ry);
590
591 let mut verifier_transcript = Transcript::new(b"example");
592 assert!(proof
593 .verify(
594 inst.get_num_vars(),
595 inst.get_num_cons(),
596 &input,
597 &inst_evals,
598 &mut verifier_transcript,
599 &gens,
600 )
601 .is_ok());
602 }
603}