sonobe/folding-schemes/src/folding/nova/zk.rs
folding-schemes/src/folding/nova/zk.rsRaw
1/// Implements Nova's zero-knowledge layer, as described in https://eprint.iacr.org/2023/573.pdf.
2///
3/// Remark: this zk layer implementation only covers a subset of the use cases:
4///
5/// We identify 3 interesting places to use the nova zk-layer: one before all the folding pipeline
6/// (Use-case-1), one at the end of the folding pipeline right before the final Decider SNARK
7/// proof (Use-case-2), and a third one for cases where compressed SNARK proofs are not needed, and
8/// just IVC proofs (bigger than SNARK proofs) suffice (Use-case-3):
9///
10/// * Use-case-1: at the beginning of the folding pipeline, right when the user has their original
11/// instance prior to be folded into the running instance, the user can fold it with the
12/// random-satisfying-instance to then have a blinded instance that can be sent to a server that
13/// will fold it with the running instance.
14/// --> In this one, the user could externalize all the IVC folding and also the Decider
15/// final proof generation to a server.
16/// * Use-case-2: at the end of all the IVC folding steps (after n iterations of nova.prove_step),
17/// to 'blind' the IVC proof so then it can be sent to a server that will generate the final
18/// decider snark proof.
19/// --> In this one, the user could externalize the Decider final proof generation to a
20/// server.
21/// * Use-case-3: the user does not care about the Decider (final compressed SNARK proof), and
22/// wants to generate a zk-proof of the IVC state to an IVC verifier (without any SNARK proof
23/// involved). Note that this proof will be much bigger and expensive to verify than a Decider
24/// SNARK proof.
25///
26/// The current implementation covers the Use-case-3.
27/// Use-case-1 can be achieved directly by a simpler version of the zk IVC scheme skipping steps
28/// and implemented directly at the app level by folding the original instance with a randomized
29/// instance (steps 2,3,4 from section D.4 of the [HyperNova](https://eprint.iacr.org/2023/573.pdf)
30/// paper).
31/// And the Use-case-2 would require a modified version of the Decider circuits.
32///
33use ark_crypto_primitives::sponge::CryptographicSponge;
34use ark_ff::{BigInteger, PrimeField};
35use ark_std::{One, Zero};
36
37use crate::{
38 arith::r1cs::{RelaxedR1CS, R1CS},
39 folding::traits::CommittedInstanceOps,
40 RngCore,
41};
42use ark_crypto_primitives::sponge::{
43 poseidon::{PoseidonConfig, PoseidonSponge},
44 Absorb,
45};
46use ark_ec::{CurveGroup, Group};
47use ark_r1cs_std::{
48 groups::{CurveVar, GroupOpsBounds},
49 ToConstraintFieldGadget,
50};
51
52use crate::{commitment::CommitmentScheme, folding::circuits::CF2, frontend::FCircuit, Error};
53
54use super::{circuits::ChallengeGadget, nifs::NIFS, CommittedInstance, Nova, Witness};
55
56// We use the same definition of a folding proof as in https://eprint.iacr.org/2023/969.pdf
57// It consists in the commitment to the T term
58pub struct FoldingProof<C: CurveGroup> {
59 cmT: C,
60}
61
62pub struct RandomizedIVCProof<C1: CurveGroup, C2: CurveGroup> {
63 pub U_i: CommittedInstance<C1>,
64 pub u_i: CommittedInstance<C1>,
65 pub U_r: CommittedInstance<C1>,
66 pub pi: FoldingProof<C1>,
67 pub pi_prime: FoldingProof<C1>,
68 pub W_i_prime: Witness<C1>,
69 pub cf_U_i: CommittedInstance<C2>,
70 pub cf_W_i: Witness<C2>,
71}
72
73impl<C1: CurveGroup, C2: CurveGroup> RandomizedIVCProof<C1, C2>
74where
75 <C1 as Group>::ScalarField: Absorb,
76 <C1 as CurveGroup>::BaseField: PrimeField,
77{
78 /// Computes challenge required before folding instances
79 fn get_folding_challenge(
80 sponge: &mut PoseidonSponge<C1::ScalarField>,
81 pp_hash: C1::ScalarField,
82 U_i: CommittedInstance<C1>,
83 u_i: CommittedInstance<C1>,
84 cmT: C1,
85 ) -> Result<C1::ScalarField, Error> {
86 let r_bits = ChallengeGadget::<C1>::get_challenge_native(sponge, pp_hash, U_i, u_i, cmT);
87 C1::ScalarField::from_bigint(BigInteger::from_bits_le(&r_bits)).ok_or(Error::OutOfBounds)
88 }
89
90 /// Compute a zero-knowledge proof of a Nova IVC proof
91 /// It implements the prover of appendix D.4.in https://eprint.iacr.org/2023/573.pdf
92 /// For further details on why folding is hiding, see lemma 9
93 pub fn new<
94 GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
95 GC2: CurveVar<C2, CF2<C2>>,
96 FC: FCircuit<C1::ScalarField>,
97 CS1: CommitmentScheme<C1, true>,
98 CS2: CommitmentScheme<C2, true>,
99 >(
100 nova: &Nova<C1, GC1, C2, GC2, FC, CS1, CS2, true>,
101 mut rng: impl RngCore,
102 ) -> Result<RandomizedIVCProof<C1, C2>, Error>
103 where
104 <C1 as Group>::ScalarField: Absorb,
105 <C2 as Group>::ScalarField: Absorb,
106 <C2 as Group>::ScalarField: PrimeField,
107 <C2 as CurveGroup>::BaseField: PrimeField,
108 <C2 as CurveGroup>::BaseField: Absorb,
109 for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
110 GC2: ToConstraintFieldGadget<<C2 as CurveGroup>::BaseField>,
111 C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
112 {
113 let mut challenges_sponge = PoseidonSponge::<C1::ScalarField>::new(&nova.poseidon_config);
114
115 // I. Compute proof for 'regular' instances
116 // 1. Fold the instance-witness pairs (U_i, W_i) with (u_i, w_i)
117 // a. Compute T
118 let (T, cmT) = NIFS::<C1, CS1, true>::compute_cmT(
119 &nova.cs_pp,
120 &nova.r1cs,
121 &nova.w_i,
122 &nova.u_i,
123 &nova.W_i,
124 &nova.U_i,
125 )?;
126
127 // b. Compute folding challenge
128 let r = RandomizedIVCProof::<C1, C2>::get_folding_challenge(
129 &mut challenges_sponge,
130 nova.pp_hash,
131 nova.U_i.clone(),
132 nova.u_i.clone(),
133 cmT,
134 )?;
135
136 // c. Compute fold
137 let (W_f, U_f) = NIFS::<C1, CS1, true>::fold_instances(
138 r, &nova.w_i, &nova.u_i, &nova.W_i, &nova.U_i, &T, cmT,
139 )?;
140
141 // d. Store folding proof
142 let pi = FoldingProof { cmT };
143
144 // 2. Sample a satisfying relaxed R1CS instance-witness pair (W_r, U_r)
145 let (W_r, U_r) = nova.r1cs.sample::<CS1>(&nova.cs_pp, &mut rng)?;
146
147 // 3. Fold the instance-witness pair (U_f, W_f) with (U_r, W_r)
148 // a. Compute T
149 let (T_i_prime, cmT_i_prime) =
150 NIFS::<C1, CS1, true>::compute_cmT(&nova.cs_pp, &nova.r1cs, &W_f, &U_f, &W_r, &U_r)?;
151
152 // b. Compute folding challenge
153 let r_2 = RandomizedIVCProof::<C1, C2>::get_folding_challenge(
154 &mut challenges_sponge,
155 nova.pp_hash,
156 U_f.clone(),
157 U_r.clone(),
158 cmT_i_prime,
159 )?;
160
161 // c. Compute fold
162 let (W_i_prime, _) = NIFS::<C1, CS1, true>::fold_instances(
163 r_2,
164 &W_f,
165 &U_f,
166 &W_r,
167 &U_r,
168 &T_i_prime,
169 cmT_i_prime,
170 )?;
171
172 // d. Store folding proof
173 let pi_prime = FoldingProof { cmT: cmT_i_prime };
174
175 Ok(RandomizedIVCProof {
176 U_i: nova.U_i.clone(),
177 u_i: nova.u_i.clone(),
178 U_r,
179 pi,
180 pi_prime,
181 W_i_prime,
182 cf_U_i: nova.cf_U_i.clone(),
183 cf_W_i: nova.cf_W_i.clone(),
184 })
185 }
186
187 /// Verify a zero-knowledge proof of a Nova IVC proof
188 /// It implements the verifier of appendix D.4. in https://eprint.iacr.org/2023/573.pdf
189 #[allow(clippy::too_many_arguments)]
190 pub fn verify<
191 CS1: CommitmentScheme<C1, true>,
192 GC2: CurveVar<C2, CF2<C2>>,
193 CS2: CommitmentScheme<C2, true>,
194 >(
195 r1cs: &R1CS<C1::ScalarField>,
196 cf_r1cs: &R1CS<C2::ScalarField>,
197 pp_hash: C1::ScalarField,
198 poseidon_config: &PoseidonConfig<C1::ScalarField>,
199 i: C1::ScalarField,
200 z_0: Vec<C1::ScalarField>,
201 z_i: Vec<C1::ScalarField>,
202 proof: &RandomizedIVCProof<C1, C2>,
203 ) -> Result<(), Error>
204 where
205 <C1 as Group>::ScalarField: Absorb,
206 <C2 as Group>::ScalarField: Absorb,
207 <C2 as CurveGroup>::BaseField: PrimeField,
208 <C2 as CurveGroup>::BaseField: Absorb,
209 for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
210 GC2: ToConstraintFieldGadget<<C2 as CurveGroup>::BaseField>,
211 C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
212 {
213 // Handles case where i=0
214 if i == C1::ScalarField::zero() {
215 if z_0 == z_i {
216 return Ok(());
217 } else {
218 return Err(Error::zkIVCVerificationFail);
219 }
220 }
221
222 // 1. Check that u_i.x is correct - including the cyclefold running instance
223 // a. Check length
224 if proof.u_i.x.len() != 2 {
225 return Err(Error::IVCVerificationFail);
226 }
227
228 // b. Check computed hashes are correct
229 let mut sponge = PoseidonSponge::<C1::ScalarField>::new(poseidon_config);
230 let expected_u_i_x = proof.U_i.hash(&sponge, pp_hash, i, &z_0, &z_i);
231 if expected_u_i_x != proof.u_i.x[0] {
232 return Err(Error::zkIVCVerificationFail);
233 }
234
235 let expected_cf_u_i_x = proof.cf_U_i.hash_cyclefold(&sponge, pp_hash);
236 if expected_cf_u_i_x != proof.u_i.x[1] {
237 return Err(Error::IVCVerificationFail);
238 }
239
240 // 2. Check that u_i values are correct
241 if !proof.u_i.cmE.is_zero() || proof.u_i.u != C1::ScalarField::one() {
242 return Err(Error::zkIVCVerificationFail);
243 }
244
245 // 3. Obtain the U_f folded instance
246 // a. Compute folding challenge
247 let r = RandomizedIVCProof::<C1, C2>::get_folding_challenge(
248 &mut sponge,
249 pp_hash,
250 proof.U_i.clone(),
251 proof.u_i.clone(),
252 proof.pi.cmT,
253 )?;
254
255 // b. Get the U_f instance
256 let U_f = NIFS::<C1, CS1, true>::fold_committed_instance(
257 r,
258 &proof.u_i,
259 &proof.U_i,
260 &proof.pi.cmT,
261 );
262
263 // 4. Obtain the U^{\prime}_i folded instance
264 // a. Compute folding challenge
265 let r_2 = RandomizedIVCProof::<C1, C2>::get_folding_challenge(
266 &mut sponge,
267 pp_hash,
268 U_f.clone(),
269 proof.U_r.clone(),
270 proof.pi_prime.cmT,
271 )?;
272
273 // b. Compute fold
274 let U_i_prime = NIFS::<C1, CS1, true>::fold_committed_instance(
275 r_2,
276 &U_f,
277 &proof.U_r,
278 &proof.pi_prime.cmT,
279 );
280
281 // 5. Check that W^{\prime}_i is a satisfying witness
282 r1cs.check_relaxed_relation(&proof.W_i_prime, &U_i_prime)?;
283
284 // 6. Check that the cyclefold instance-witness pair satisfies the cyclefold relaxed r1cs
285 cf_r1cs.check_relaxed_relation(&proof.cf_W_i, &proof.cf_U_i)?;
286
287 Ok(())
288 }
289}
290
291#[cfg(test)]
292pub mod tests {
293 use super::*;
294 use crate::commitment::pedersen::Pedersen;
295 use crate::folding::nova::tests::test_ivc_opt;
296 use crate::frontend::utils::CubicFCircuit;
297 use crate::transcript::poseidon::poseidon_canonical_config;
298 use ark_bn254::{Fr, G1Projective as Projective};
299 use ark_grumpkin::{constraints::GVar as GVar2, Projective as Projective2};
300 use rand::rngs::OsRng;
301
302 // Tests zk proof generation and verification for a valid nova IVC proof
303 #[test]
304 fn test_zk_nova_ivc() {
305 let mut rng = OsRng;
306 let poseidon_config = poseidon_canonical_config::<Fr>();
307 let F_circuit = CubicFCircuit::<Fr>::new(()).unwrap();
308 let (_, nova) = test_ivc_opt::<Pedersen<Projective, true>, Pedersen<Projective2, true>, true>(
309 poseidon_config.clone(),
310 F_circuit,
311 3,
312 );
313
314 let proof = RandomizedIVCProof::new(&nova, &mut rng).unwrap();
315 let verify = RandomizedIVCProof::verify::<
316 Pedersen<Projective, true>,
317 GVar2,
318 Pedersen<Projective2, true>,
319 >(
320 &nova.r1cs,
321 &nova.cf_r1cs,
322 nova.pp_hash,
323 &nova.poseidon_config,
324 nova.i,
325 nova.z_0,
326 nova.z_i,
327 &proof,
328 );
329 assert!(verify.is_ok());
330 }
331
332 #[test]
333 fn test_zk_nova_when_i_is_zero() {
334 let mut rng = OsRng;
335 let poseidon_config = poseidon_canonical_config::<Fr>();
336 let F_circuit = CubicFCircuit::<Fr>::new(()).unwrap();
337 let (_, nova) = test_ivc_opt::<Pedersen<Projective, true>, Pedersen<Projective2, true>, true>(
338 poseidon_config.clone(),
339 F_circuit,
340 0,
341 );
342
343 let proof = RandomizedIVCProof::new(&nova, &mut rng).unwrap();
344 let verify = RandomizedIVCProof::verify::<
345 Pedersen<Projective, true>,
346 GVar2,
347 Pedersen<Projective2, true>,
348 >(
349 &nova.r1cs,
350 &nova.cf_r1cs,
351 nova.pp_hash,
352 &nova.poseidon_config,
353 nova.i,
354 nova.z_0,
355 nova.z_i,
356 &proof,
357 );
358 assert!(verify.is_ok());
359 }
360
361 #[test]
362 fn test_zk_nova_verification_fails_with_wrong_running_instance() {
363 let mut rng = OsRng;
364 let poseidon_config = poseidon_canonical_config::<Fr>();
365 let F_circuit = CubicFCircuit::<Fr>::new(()).unwrap();
366 let (_, nova) = test_ivc_opt::<Pedersen<Projective, true>, Pedersen<Projective2, true>, true>(
367 poseidon_config.clone(),
368 F_circuit,
369 3,
370 );
371 let (_, sampled_committed_instance) = nova
372 .r1cs
373 .sample::<Pedersen<Projective, true>>(&nova.cs_pp, rng)
374 .unwrap();
375
376 // proof verification fails with incorrect running instance
377 let mut nova_with_incorrect_running_instance = nova.clone();
378 nova_with_incorrect_running_instance.U_i = sampled_committed_instance;
379 let incorrect_proof =
380 RandomizedIVCProof::new(&nova_with_incorrect_running_instance, &mut rng).unwrap();
381 let verify = RandomizedIVCProof::verify::<
382 Pedersen<Projective, true>,
383 GVar2,
384 Pedersen<Projective2, true>,
385 >(
386 &nova_with_incorrect_running_instance.r1cs,
387 &nova_with_incorrect_running_instance.cf_r1cs,
388 nova_with_incorrect_running_instance.pp_hash,
389 &nova_with_incorrect_running_instance.poseidon_config,
390 nova_with_incorrect_running_instance.i,
391 nova_with_incorrect_running_instance.z_0,
392 nova_with_incorrect_running_instance.z_i,
393 &incorrect_proof,
394 );
395 assert!(verify.is_err());
396 }
397
398 #[test]
399 fn test_zk_nova_verification_fails_with_wrong_running_witness() {
400 let mut rng = OsRng;
401 let poseidon_config = poseidon_canonical_config::<Fr>();
402 let F_circuit = CubicFCircuit::<Fr>::new(()).unwrap();
403 let (_, nova) = test_ivc_opt::<Pedersen<Projective, true>, Pedersen<Projective2, true>, true>(
404 poseidon_config.clone(),
405 F_circuit,
406 3,
407 );
408 let (sampled_committed_witness, _) = nova
409 .r1cs
410 .sample::<Pedersen<Projective, true>>(&nova.cs_pp, rng)
411 .unwrap();
412
413 // proof generation fails with incorrect running witness
414 let mut nova_with_incorrect_running_witness = nova.clone();
415 nova_with_incorrect_running_witness.W_i = sampled_committed_witness;
416 let incorrect_proof =
417 RandomizedIVCProof::new(&nova_with_incorrect_running_witness, &mut rng).unwrap();
418 let verify = RandomizedIVCProof::verify::<
419 Pedersen<Projective, true>,
420 GVar2,
421 Pedersen<Projective2, true>,
422 >(
423 &nova_with_incorrect_running_witness.r1cs,
424 &nova_with_incorrect_running_witness.cf_r1cs,
425 nova_with_incorrect_running_witness.pp_hash,
426 &nova_with_incorrect_running_witness.poseidon_config,
427 nova_with_incorrect_running_witness.i,
428 nova_with_incorrect_running_witness.z_0,
429 nova_with_incorrect_running_witness.z_i,
430 &incorrect_proof,
431 );
432 assert!(verify.is_err());
433 }
434}