Nova/src/lib.rs
src/lib.rsRaw
1//! This library implements core components of Nova.
2#![allow(non_snake_case)]
3#![allow(clippy::type_complexity)]
4#![deny(missing_docs)]
5
6mod commitments;
7
8pub mod bellperson;
9pub mod errors;
10pub mod pasta;
11pub mod r1cs;
12pub mod traits;
13
14use std::marker::PhantomData;
15
16use commitments::{AppendToTranscriptTrait, CompressedCommitment};
17use errors::NovaError;
18use merlin::Transcript;
19use r1cs::{
20 R1CSGens, R1CSInstance, R1CSShape, R1CSWitness, RelaxedR1CSInstance, RelaxedR1CSWitness,
21};
22use traits::{ChallengeTrait, Group};
23
24/// A SNARK that holds the proof of a step of an incremental computation
25pub struct StepSNARK<G: Group> {
26 comm_T: CompressedCommitment<G::CompressedGroupElement>,
27 _p: PhantomData<G>,
28}
29
30impl<G: Group> StepSNARK<G> {
31 fn protocol_name() -> &'static [u8] {
32 b"NovaStepSNARK"
33 }
34
35 /// Takes as input a Relaxed R1CS instance-witness tuple `(U1, W1)` and
36 /// an R1CS instance-witness tuple `(U2, W2)` with the same structure `shape`
37 /// and defined with respect to the same `gens`, and outputs
38 /// a folded Relaxed R1CS instance-witness tuple `(U, W)` of the same shape `shape`,
39 /// with the guarantee that the folded witness `W` satisfies the folded instance `U`
40 /// if and only if `W1` satisfies `U1` and `W2` satisfies `U2`.
41 pub fn prove(
42 gens: &R1CSGens<G>,
43 S: &R1CSShape<G>,
44 U1: &RelaxedR1CSInstance<G>,
45 W1: &RelaxedR1CSWitness<G>,
46 U2: &R1CSInstance<G>,
47 W2: &R1CSWitness<G>,
48 transcript: &mut Transcript,
49 ) -> Result<
50 (
51 StepSNARK<G>,
52 (RelaxedR1CSInstance<G>, RelaxedR1CSWitness<G>),
53 ),
54 NovaError,
55 > {
56 // append the protocol name to the transcript
57 //transcript.append_protocol_name(StepSNARK::protocol_name());
58 transcript.append_message(b"protocol-name", StepSNARK::<G>::protocol_name());
59
60 // compute a commitment to the cross-term
61 let (T, comm_T) = S.commit_T(gens, U1, W1, U2, W2)?;
62
63 // append `comm_T` to the transcript and obtain a challenge
64 comm_T.append_to_transcript(b"comm_T", transcript);
65
66 // compute a challenge from the transcript
67 let r = G::Scalar::challenge(b"r", transcript);
68
69 // fold the instance using `r` and `comm_T`
70 let U = U1.fold(U2, &comm_T, &r)?;
71
72 // fold the witness using `r` and `T`
73 let W = W1.fold(W2, &T, &r)?;
74
75 // return the folded instance and witness
76 Ok((
77 StepSNARK {
78 comm_T,
79 _p: Default::default(),
80 },
81 (U, W),
82 ))
83 }
84
85 /// Takes as input a relaxed R1CS instance `U1` and and R1CS instance `U2`
86 /// with the same shape and defined with respect to the same parameters,
87 /// and outputs a folded instance `U` with the same shape,
88 /// with the guarantee that the folded instance `U`
89 /// if and only if `U1` and `U2` are satisfiable.
90 pub fn verify(
91 &self,
92 U1: &RelaxedR1CSInstance<G>,
93 U2: &R1CSInstance<G>,
94 transcript: &mut Transcript,
95 ) -> Result<RelaxedR1CSInstance<G>, NovaError> {
96 // append the protocol name to the transcript
97 transcript.append_message(b"protocol-name", StepSNARK::<G>::protocol_name());
98
99 // append `comm_T` to the transcript and obtain a challenge
100 self.comm_T.append_to_transcript(b"comm_T", transcript);
101
102 // compute a challenge from the transcript
103 let r = G::Scalar::challenge(b"r", transcript);
104
105 // fold the instance using `r` and `comm_T`
106 let U = U1.fold(U2, &self.comm_T, &r)?;
107
108 // return the folded instance
109 Ok(U)
110 }
111}
112
113/// A SNARK that holds the proof of the final step of an incremental computation
114pub struct FinalSNARK<G: Group> {
115 W: RelaxedR1CSWitness<G>,
116}
117
118impl<G: Group> FinalSNARK<G> {
119 /// Produces a proof of a instance given its satisfying witness `W`.
120 pub fn prove(W: &RelaxedR1CSWitness<G>) -> Result<FinalSNARK<G>, NovaError> {
121 Ok(FinalSNARK { W: W.clone() })
122 }
123
124 /// Verifies the proof of a folded instance `U` given its shape `S` public parameters `gens`
125 pub fn verify(
126 &self,
127 gens: &R1CSGens<G>,
128 S: &R1CSShape<G>,
129 U: &RelaxedR1CSInstance<G>,
130 ) -> Result<(), NovaError> {
131 // check that the witness is a valid witness to the folded instance `U`
132 S.is_sat_relaxed(gens, U, &self.W)
133 }
134}
135
136#[cfg(test)]
137mod tests {
138 use super::*;
139 use crate::traits::PrimeField;
140 use rand::rngs::OsRng;
141
142 type S = pasta_curves::pallas::Scalar;
143 type G = pasta_curves::pallas::Point;
144
145 #[test]
146 fn test_tiny_r1cs() {
147 let one = S::one();
148 let (num_cons, num_vars, num_io, A, B, C) = {
149 let num_cons = 4;
150 let num_vars = 4;
151 let num_io = 2;
152
153 // Consider a cubic equation: `x^3 + x + 5 = y`, where `x` and `y` are respectively the input and output.
154 // The R1CS for this problem consists of the following constraints:
155 // `I0 * I0 - Z0 = 0`
156 // `Z0 * I0 - Z1 = 0`
157 // `(Z1 + I0) * 1 - Z2 = 0`
158 // `(Z2 + 5) * 1 - I1 = 0`
159
160 // Relaxed R1CS is a set of three sparse matrices (A B C), where there is a row for every
161 // constraint and a column for every entry in z = (vars, u, inputs)
162 // An R1CS instance is satisfiable iff:
163 // Az \circ Bz = u \cdot Cz + E, where z = (vars, 1, inputs)
164 let mut A: Vec<(usize, usize, S)> = Vec::new();
165 let mut B: Vec<(usize, usize, S)> = Vec::new();
166 let mut C: Vec<(usize, usize, S)> = Vec::new();
167
168 // constraint 0 entries in (A,B,C)
169 // `I0 * I0 - Z0 = 0`
170 A.push((0, num_vars + 1, one));
171 B.push((0, num_vars + 1, one));
172 C.push((0, 0, one));
173
174 // constraint 1 entries in (A,B,C)
175 // `Z0 * I0 - Z1 = 0`
176 A.push((1, 0, one));
177 B.push((1, num_vars + 1, one));
178 C.push((1, 1, one));
179
180 // constraint 2 entries in (A,B,C)
181 // `(Z1 + I0) * 1 - Z2 = 0`
182 A.push((2, 1, one));
183 A.push((2, num_vars + 1, one));
184 B.push((2, num_vars, one));
185 C.push((2, 2, one));
186
187 // constraint 3 entries in (A,B,C)
188 // `(Z2 + 5) * 1 - I1 = 0`
189 A.push((3, 2, one));
190 A.push((3, num_vars, one + one + one + one + one));
191 B.push((3, num_vars, one));
192 C.push((3, num_vars + 2, one));
193
194 (num_cons, num_vars, num_io, A, B, C)
195 };
196
197 // create a shape object
198 let S = {
199 let res = R1CSShape::new(num_cons, num_vars, num_io, &A, &B, &C);
200 assert!(res.is_ok());
201 res.unwrap()
202 };
203
204 // generate generators
205 let gens = R1CSGens::new(num_cons, num_vars);
206
207 let rand_inst_witness_generator =
208 |gens: &R1CSGens<G>, I: &S| -> (S, R1CSInstance<G>, R1CSWitness<G>) {
209 let i0 = *I;
210
211 // compute a satisfying (vars, X) tuple
212 let (O, vars, X) = {
213 let z0 = i0 * i0; // constraint 0
214 let z1 = i0 * z0; // constraint 1
215 let z2 = z1 + i0; // constraint 2
216 let i1 = z2 + one + one + one + one + one; // constraint 3
217
218 // store the witness and IO for the instance
219 let W = vec![z0, z1, z2, S::zero()];
220 let X = vec![i0, i1];
221 (i1, W, X)
222 };
223
224 let W = {
225 let res = R1CSWitness::new(&S, &vars);
226 assert!(res.is_ok());
227 res.unwrap()
228 };
229 let U = {
230 let comm_W = W.commit(gens);
231 let res = R1CSInstance::new(&S, &comm_W, &X);
232 assert!(res.is_ok());
233 res.unwrap()
234 };
235
236 // check that generated instance is satisfiable
237 assert!(S.is_sat(gens, &U, &W).is_ok());
238
239 (O, U, W)
240 };
241
242 let mut csprng: OsRng = OsRng;
243 let I = S::random(&mut csprng); // the first input is picked randomly for the first instance
244 let (O, U1, W1) = rand_inst_witness_generator(&gens, &I);
245 let (_O, U2, W2) = rand_inst_witness_generator(&gens, &O);
246
247 // produce a default running instance
248 let mut r_W = RelaxedR1CSWitness::default(&S);
249 let mut r_U = RelaxedR1CSInstance::default(&gens, &S);
250
251 // produce a step SNARK with (W1, U1) as the first incoming witness-instance pair
252 let mut prover_transcript = Transcript::new(b"StepSNARKExample");
253 let res = StepSNARK::prove(&gens, &S, &r_U, &r_W, &U1, &W1, &mut prover_transcript);
254 assert!(res.is_ok());
255 let (step_snark, (_U, W)) = res.unwrap();
256
257 // verify the step SNARK with U1 as the first incoming instance
258 let mut verifier_transcript = Transcript::new(b"StepSNARKExample");
259 let res = step_snark.verify(&r_U, &U1, &mut verifier_transcript);
260 assert!(res.is_ok());
261 let U = res.unwrap();
262
263 assert_eq!(U, _U);
264
265 // update the running witness and instance
266 r_W = W;
267 r_U = U;
268
269 // produce a step SNARK with (W2, U2) as the second incoming witness-instance pair
270 let res = StepSNARK::prove(&gens, &S, &r_U, &r_W, &U2, &W2, &mut prover_transcript);
271 assert!(res.is_ok());
272 let (step_snark, (_U, W)) = res.unwrap();
273
274 // verify the step SNARK with U1 as the first incoming instance
275 let res = step_snark.verify(&r_U, &U2, &mut verifier_transcript);
276 assert!(res.is_ok());
277 let U = res.unwrap();
278
279 assert_eq!(U, _U);
280
281 // update the running witness and instance
282 r_W = W;
283 r_U = U;
284
285 // produce a final SNARK
286 let res = FinalSNARK::prove(&r_W);
287 assert!(res.is_ok());
288 let final_snark = res.unwrap();
289 // verify the final SNARK
290 let res = final_snark.verify(&gens, &S, &r_U);
291 assert!(res.is_ok());
292 }
293}