Nova/src/spartan_with_ipa_pc/mod.rs
src/spartan_with_ipa_pc/mod.rsRaw
1//! This module implements RelaxedR1CSSNARKTrait using a Spartan variant
2//! instantiated with an IPA-based polynomial commitment scheme
3mod ipa;
4mod polynomial;
5mod sumcheck;
6
7use super::{
8 commitments::CommitGens,
9 errors::NovaError,
10 r1cs::{R1CSGens, R1CSShape, RelaxedR1CSInstance, RelaxedR1CSWitness},
11 traits::{
12 snark::{ProverKeyTrait, RelaxedR1CSSNARKTrait, VerifierKeyTrait},
13 AppendToTranscriptTrait, ChallengeTrait, Group,
14 },
15};
16use core::cmp::max;
17use ff::Field;
18use ipa::{InnerProductArgument, InnerProductInstance, InnerProductWitness, NIFSForInnerProduct};
19use itertools::concat;
20use merlin::Transcript;
21use polynomial::{EqPolynomial, MultilinearPolynomial, SparsePolynomial};
22use rayon::prelude::*;
23use sumcheck::SumcheckProof;
24
25/// A type that represents the prover's key
26pub struct ProverKey<G: Group> {
27 gens_r1cs: R1CSGens<G>,
28 gens_ipa: CommitGens<G>,
29 S: R1CSShape<G>,
30}
31
32impl<G: Group> ProverKeyTrait<G> for ProverKey<G> {
33 fn new(gens: &R1CSGens<G>, S: &R1CSShape<G>) -> Self {
34 ProverKey {
35 gens_r1cs: gens.clone(),
36 gens_ipa: CommitGens::new(b"ipa", 1),
37 S: S.clone(),
38 }
39 }
40}
41
42/// A type that represents the verifier's key
43pub struct VerifierKey<G: Group> {
44 gens_r1cs: R1CSGens<G>,
45 gens_ipa: CommitGens<G>,
46 S: R1CSShape<G>,
47}
48
49impl<G: Group> VerifierKeyTrait<G> for VerifierKey<G> {
50 fn new(gens: &R1CSGens<G>, S: &R1CSShape<G>) -> Self {
51 VerifierKey {
52 gens_r1cs: gens.clone(),
53 gens_ipa: CommitGens::new(b"ipa", 1),
54 S: S.clone(),
55 }
56 }
57}
58
59/// A succinct proof of knowledge of a witness to a relaxed R1CS instance
60/// The proof is produced using Spartan's combination of the sum-check and
61/// the commitment to a vector viewed as a polynomial commitment
62pub struct RelaxedR1CSSNARK<G: Group> {
63 sc_proof_outer: SumcheckProof<G>,
64 claims_outer: (G::Scalar, G::Scalar, G::Scalar),
65 sc_proof_inner: SumcheckProof<G>,
66 eval_E: G::Scalar,
67 eval_W: G::Scalar,
68 nifs_ip: NIFSForInnerProduct<G>,
69 ipa: InnerProductArgument<G>,
70}
71
72impl<G: Group> RelaxedR1CSSNARKTrait<G> for RelaxedR1CSSNARK<G> {
73 type ProverKey = ProverKey<G>;
74 type VerifierKey = VerifierKey<G>;
75
76 /// produces a succinct proof of satisfiability of a RelaxedR1CS instance
77 fn prove(
78 pk: &Self::ProverKey,
79 U: &RelaxedR1CSInstance<G>,
80 W: &RelaxedR1CSWitness<G>,
81 ) -> Result<Self, NovaError> {
82 let mut transcript = Transcript::new(b"RelaxedR1CSSNARK");
83
84 debug_assert!(pk.S.is_sat_relaxed(&pk.gens_r1cs, U, W).is_ok());
85
86 // sanity check that R1CSShape has certain size characteristics
87 assert_eq!(pk.S.num_cons.next_power_of_two(), pk.S.num_cons);
88 assert_eq!(pk.S.num_vars.next_power_of_two(), pk.S.num_vars);
89 assert_eq!(pk.S.num_io.next_power_of_two(), pk.S.num_io);
90 assert!(pk.S.num_io < pk.S.num_vars);
91
92 // append the R1CSShape and RelaxedR1CSInstance to the transcript
93 pk.S.append_to_transcript(b"S", &mut transcript);
94 U.append_to_transcript(b"U", &mut transcript);
95
96 // compute the full satisfying assignment by concatenating W.W, U.u, and U.X
97 let mut z = concat(vec![W.W.clone(), vec![U.u], U.X.clone()]);
98
99 let (num_rounds_x, num_rounds_y) = (
100 (pk.S.num_cons as f64).log2() as usize,
101 ((pk.S.num_vars as f64).log2() as usize + 1),
102 );
103
104 // outer sum-check
105 let tau = (0..num_rounds_x)
106 .map(|_i| G::Scalar::challenge(b"challenge_tau", &mut transcript))
107 .collect();
108
109 let mut poly_tau = MultilinearPolynomial::new(EqPolynomial::new(tau).evals());
110 let (mut poly_Az, mut poly_Bz, poly_Cz, mut poly_uCz_E) = {
111 let (poly_Az, poly_Bz, poly_Cz) = pk.S.multiply_vec(&z)?;
112 let poly_uCz_E = (0..pk.S.num_cons)
113 .map(|i| U.u * poly_Cz[i] + W.E[i])
114 .collect::<Vec<G::Scalar>>();
115 (
116 MultilinearPolynomial::new(poly_Az),
117 MultilinearPolynomial::new(poly_Bz),
118 MultilinearPolynomial::new(poly_Cz),
119 MultilinearPolynomial::new(poly_uCz_E),
120 )
121 };
122
123 let comb_func_outer =
124 |poly_A_comp: &G::Scalar,
125 poly_B_comp: &G::Scalar,
126 poly_C_comp: &G::Scalar,
127 poly_D_comp: &G::Scalar|
128 -> G::Scalar { *poly_A_comp * (*poly_B_comp * *poly_C_comp - *poly_D_comp) };
129 let (sc_proof_outer, r_x, claims_outer) = SumcheckProof::prove_cubic_with_additive_term(
130 &G::Scalar::zero(), // claim is zero
131 num_rounds_x,
132 &mut poly_tau,
133 &mut poly_Az,
134 &mut poly_Bz,
135 &mut poly_uCz_E,
136 comb_func_outer,
137 &mut transcript,
138 );
139
140 // claims from the end of sum-check
141 let (claim_Az, claim_Bz): (G::Scalar, G::Scalar) = (claims_outer[1], claims_outer[2]);
142
143 claim_Az.append_to_transcript(b"claim_Az", &mut transcript);
144 claim_Bz.append_to_transcript(b"claim_Bz", &mut transcript);
145 let claim_Cz = poly_Cz.evaluate(&r_x);
146 let eval_E = MultilinearPolynomial::new(W.E.clone()).evaluate(&r_x);
147 claim_Cz.append_to_transcript(b"claim_Cz", &mut transcript);
148 eval_E.append_to_transcript(b"eval_E", &mut transcript);
149
150 // inner sum-check
151 let r_A = G::Scalar::challenge(b"challenge_rA", &mut transcript);
152 let r_B = G::Scalar::challenge(b"challenge_rB", &mut transcript);
153 let r_C = G::Scalar::challenge(b"challenge_rC", &mut transcript);
154 let claim_inner_joint = r_A * claim_Az + r_B * claim_Bz + r_C * claim_Cz;
155
156 let poly_ABC = {
157 // compute the initial evaluation table for R(\tau, x)
158 let evals_rx = EqPolynomial::new(r_x.clone()).evals();
159
160 // Bounds "row" variables of (A, B, C) matrices viewed as 2d multilinear polynomials
161 let compute_eval_table_sparse =
162 |S: &R1CSShape<G>, rx: &[G::Scalar]| -> (Vec<G::Scalar>, Vec<G::Scalar>, Vec<G::Scalar>) {
163 assert_eq!(rx.len(), S.num_cons);
164
165 let inner = |M: &Vec<(usize, usize, G::Scalar)>, M_evals: &mut Vec<G::Scalar>| {
166 for (row, col, val) in M {
167 M_evals[*col] += rx[*row] * val;
168 }
169 };
170
171 let (A_evals, (B_evals, C_evals)) = rayon::join(
172 || {
173 let mut A_evals: Vec<G::Scalar> = vec![G::Scalar::zero(); 2 * S.num_vars];
174 inner(&S.A, &mut A_evals);
175 A_evals
176 },
177 || {
178 rayon::join(
179 || {
180 let mut B_evals: Vec<G::Scalar> = vec![G::Scalar::zero(); 2 * S.num_vars];
181 inner(&S.B, &mut B_evals);
182 B_evals
183 },
184 || {
185 let mut C_evals: Vec<G::Scalar> = vec![G::Scalar::zero(); 2 * S.num_vars];
186 inner(&S.C, &mut C_evals);
187 C_evals
188 },
189 )
190 },
191 );
192
193 (A_evals, B_evals, C_evals)
194 };
195
196 let (evals_A, evals_B, evals_C) = compute_eval_table_sparse(&pk.S, &evals_rx);
197
198 assert_eq!(evals_A.len(), evals_B.len());
199 assert_eq!(evals_A.len(), evals_C.len());
200 (0..evals_A.len())
201 .into_par_iter()
202 .map(|i| r_A * evals_A[i] + r_B * evals_B[i] + r_C * evals_C[i])
203 .collect::<Vec<G::Scalar>>()
204 };
205
206 let poly_z = {
207 z.resize(pk.S.num_vars * 2, G::Scalar::zero());
208 z
209 };
210
211 let comb_func = |poly_A_comp: &G::Scalar, poly_B_comp: &G::Scalar| -> G::Scalar {
212 *poly_A_comp * *poly_B_comp
213 };
214 let (sc_proof_inner, r_y, _claims_inner) = SumcheckProof::prove_quad(
215 &claim_inner_joint,
216 num_rounds_y,
217 &mut MultilinearPolynomial::new(poly_ABC),
218 &mut MultilinearPolynomial::new(poly_z),
219 comb_func,
220 &mut transcript,
221 );
222
223 let eval_W = MultilinearPolynomial::new(W.W.clone()).evaluate(&r_y[1..]);
224 eval_W.append_to_transcript(b"eval_W", &mut transcript);
225
226 let (nifs_ip, r_U, r_W) = NIFSForInnerProduct::prove(
227 &InnerProductInstance::new(&U.comm_E, &EqPolynomial::new(r_x).evals(), &eval_E),
228 &InnerProductWitness::new(&W.E),
229 &InnerProductInstance::new(
230 &U.comm_W,
231 &EqPolynomial::new(r_y[1..].to_vec()).evals(),
232 &eval_W,
233 ),
234 &InnerProductWitness::new(&W.W),
235 &mut transcript,
236 );
237
238 let ipa = InnerProductArgument::prove(
239 &pk.gens_r1cs.gens,
240 &pk.gens_ipa,
241 &r_U,
242 &r_W,
243 &mut transcript,
244 )?;
245
246 Ok(RelaxedR1CSSNARK {
247 sc_proof_outer,
248 claims_outer: (claim_Az, claim_Bz, claim_Cz),
249 sc_proof_inner,
250 eval_W,
251 eval_E,
252 nifs_ip,
253 ipa,
254 })
255 }
256
257 /// verifies a proof of satisfiability of a RelaxedR1CS instance
258 fn verify(&self, vk: &Self::VerifierKey, U: &RelaxedR1CSInstance<G>) -> Result<(), NovaError> {
259 let mut transcript = Transcript::new(b"RelaxedR1CSSNARK");
260
261 // append the R1CSShape and RelaxedR1CSInstance to the transcript
262 vk.S.append_to_transcript(b"S", &mut transcript);
263 U.append_to_transcript(b"U", &mut transcript);
264
265 let (num_rounds_x, num_rounds_y) = (
266 (vk.S.num_cons as f64).log2() as usize,
267 ((vk.S.num_vars as f64).log2() as usize + 1),
268 );
269
270 // outer sum-check
271 let tau = (0..num_rounds_x)
272 .map(|_i| G::Scalar::challenge(b"challenge_tau", &mut transcript))
273 .collect::<Vec<G::Scalar>>();
274
275 let (claim_outer_final, r_x) =
276 self
277 .sc_proof_outer
278 .verify(G::Scalar::zero(), num_rounds_x, 3, &mut transcript)?;
279
280 // verify claim_outer_final
281 let (claim_Az, claim_Bz, claim_Cz) = self.claims_outer;
282 let taus_bound_rx = EqPolynomial::new(tau).evaluate(&r_x);
283 let claim_outer_final_expected =
284 taus_bound_rx * (claim_Az * claim_Bz - U.u * claim_Cz - self.eval_E);
285 if claim_outer_final != claim_outer_final_expected {
286 return Err(NovaError::InvalidSumcheckProof);
287 }
288
289 self
290 .claims_outer
291 .0
292 .append_to_transcript(b"claim_Az", &mut transcript);
293 self
294 .claims_outer
295 .1
296 .append_to_transcript(b"claim_Bz", &mut transcript);
297 self
298 .claims_outer
299 .2
300 .append_to_transcript(b"claim_Cz", &mut transcript);
301 self.eval_E.append_to_transcript(b"eval_E", &mut transcript);
302
303 // inner sum-check
304 let r_A = G::Scalar::challenge(b"challenge_rA", &mut transcript);
305 let r_B = G::Scalar::challenge(b"challenge_rB", &mut transcript);
306 let r_C = G::Scalar::challenge(b"challenge_rC", &mut transcript);
307 let claim_inner_joint =
308 r_A * self.claims_outer.0 + r_B * self.claims_outer.1 + r_C * self.claims_outer.2;
309
310 let (claim_inner_final, r_y) =
311 self
312 .sc_proof_inner
313 .verify(claim_inner_joint, num_rounds_y, 2, &mut transcript)?;
314
315 // verify claim_inner_final
316 let eval_Z = {
317 let eval_X = {
318 // constant term
319 let mut poly_X = vec![(0, U.u)];
320 //remaining inputs
321 poly_X.extend(
322 (0..U.X.len())
323 .map(|i| (i + 1, U.X[i]))
324 .collect::<Vec<(usize, G::Scalar)>>(),
325 );
326 SparsePolynomial::new((vk.S.num_vars as f64).log2() as usize, poly_X).evaluate(&r_y[1..])
327 };
328 (G::Scalar::one() - r_y[0]) * self.eval_W + r_y[0] * eval_X
329 };
330
331 let evaluate_as_sparse_polynomial = |S: &R1CSShape<G>,
332 r_x: &[G::Scalar],
333 r_y: &[G::Scalar]|
334 -> (G::Scalar, G::Scalar, G::Scalar) {
335 let evaluate_with_table =
336 |M: &[(usize, usize, G::Scalar)], T_x: &[G::Scalar], T_y: &[G::Scalar]| -> G::Scalar {
337 (0..M.len())
338 .map(|i| {
339 let (row, col, val) = M[i];
340 T_x[row] * T_y[col] * val
341 })
342 .fold(G::Scalar::zero(), |acc, x| acc + x)
343 };
344
345 let T_x = EqPolynomial::new(r_x.to_vec()).evals();
346 let T_y = EqPolynomial::new(r_y.to_vec()).evals();
347 let eval_A_r = evaluate_with_table(&S.A, &T_x, &T_y);
348 let eval_B_r = evaluate_with_table(&S.B, &T_x, &T_y);
349 let eval_C_r = evaluate_with_table(&S.C, &T_x, &T_y);
350 (eval_A_r, eval_B_r, eval_C_r)
351 };
352
353 let (eval_A_r, eval_B_r, eval_C_r) = evaluate_as_sparse_polynomial(&vk.S, &r_x, &r_y);
354 let claim_inner_final_expected = (r_A * eval_A_r + r_B * eval_B_r + r_C * eval_C_r) * eval_Z;
355 if claim_inner_final != claim_inner_final_expected {
356 return Err(NovaError::InvalidSumcheckProof);
357 }
358
359 // verify eval_W and eval_E
360 self.eval_W.append_to_transcript(b"eval_W", &mut transcript); //eval_E is already in the transcript
361
362 let r_U = self.nifs_ip.verify(
363 &InnerProductInstance::new(&U.comm_E, &EqPolynomial::new(r_x).evals(), &self.eval_E),
364 &InnerProductInstance::new(
365 &U.comm_W,
366 &EqPolynomial::new(r_y[1..].to_vec()).evals(),
367 &self.eval_W,
368 ),
369 &mut transcript,
370 );
371
372 self.ipa.verify(
373 &vk.gens_r1cs.gens,
374 &vk.gens_ipa,
375 max(vk.S.num_vars, vk.S.num_cons),
376 &r_U,
377 &mut transcript,
378 )?;
379
380 Ok(())
381 }
382}