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;
9mod circuit;
10pub mod errors;
11mod gadgets;
12pub mod pasta;
13mod poseidon;
14pub mod r1cs;
15pub mod traits;
16
17use std::marker::PhantomData;
18
19use commitments::{AppendToTranscriptTrait, CompressedCommitment};
20use errors::NovaError;
21use merlin::Transcript;
22use r1cs::{
23 R1CSGens, R1CSInstance, R1CSShape, R1CSWitness, RelaxedR1CSInstance, RelaxedR1CSWitness,
24};
25use traits::{ChallengeTrait, Group};
26
27/// A SNARK that holds the proof of a step of an incremental computation
28pub struct StepSNARK<G: Group> {
29 comm_T: CompressedCommitment<G::CompressedGroupElement>,
30 _p: PhantomData<G>,
31}
32
33impl<G: Group> StepSNARK<G> {
34 fn protocol_name() -> &'static [u8] {
35 b"NovaStepSNARK"
36 }
37
38 /// Takes as input a Relaxed R1CS instance-witness tuple `(U1, W1)` and
39 /// an R1CS instance-witness tuple `(U2, W2)` with the same structure `shape`
40 /// and defined with respect to the same `gens`, and outputs
41 /// a folded Relaxed R1CS instance-witness tuple `(U, W)` of the same shape `shape`,
42 /// with the guarantee that the folded witness `W` satisfies the folded instance `U`
43 /// if and only if `W1` satisfies `U1` and `W2` satisfies `U2`.
44 pub fn prove(
45 gens: &R1CSGens<G>,
46 S: &R1CSShape<G>,
47 U1: &RelaxedR1CSInstance<G>,
48 W1: &RelaxedR1CSWitness<G>,
49 U2: &R1CSInstance<G>,
50 W2: &R1CSWitness<G>,
51 transcript: &mut Transcript,
52 ) -> Result<
53 (
54 StepSNARK<G>,
55 (RelaxedR1CSInstance<G>, RelaxedR1CSWitness<G>),
56 ),
57 NovaError,
58 > {
59 // append the protocol name to the transcript
60 transcript.append_message(b"protocol-name", StepSNARK::<G>::protocol_name());
61
62 // compute a commitment to the cross-term
63 let (T, comm_T) = S.commit_T(gens, U1, W1, U2, W2)?;
64
65 // append `comm_T` to the transcript and obtain a challenge
66 comm_T.append_to_transcript(b"comm_T", transcript);
67
68 // compute a challenge from the transcript
69 let r = G::Scalar::challenge(b"r", transcript);
70
71 // fold the instance using `r` and `comm_T`
72 let U = U1.fold(U2, &comm_T, &r)?;
73
74 // fold the witness using `r` and `T`
75 let W = W1.fold(W2, &T, &r)?;
76
77 // return the folded instance and witness
78 Ok((
79 StepSNARK {
80 comm_T,
81 _p: Default::default(),
82 },
83 (U, W),
84 ))
85 }
86
87 /// Takes as input a relaxed R1CS instance `U1` and and R1CS instance `U2`
88 /// with the same shape and defined with respect to the same parameters,
89 /// and outputs a folded instance `U` with the same shape,
90 /// with the guarantee that the folded instance `U`
91 /// if and only if `U1` and `U2` are satisfiable.
92 pub fn verify(
93 &self,
94 U1: &RelaxedR1CSInstance<G>,
95 U2: &R1CSInstance<G>,
96 transcript: &mut Transcript,
97 ) -> Result<RelaxedR1CSInstance<G>, NovaError> {
98 // append the protocol name to the transcript
99 transcript.append_message(b"protocol-name", StepSNARK::<G>::protocol_name());
100
101 // append `comm_T` to the transcript and obtain a challenge
102 self.comm_T.append_to_transcript(b"comm_T", transcript);
103
104 // compute a challenge from the transcript
105 let r = G::Scalar::challenge(b"r", transcript);
106
107 // fold the instance using `r` and `comm_T`
108 let U = U1.fold(U2, &self.comm_T, &r)?;
109
110 // return the folded instance
111 Ok(U)
112 }
113}
114
115/// A SNARK that holds the proof of the final step of an incremental computation
116pub struct FinalSNARK<G: Group> {
117 W: RelaxedR1CSWitness<G>,
118}
119
120impl<G: Group> FinalSNARK<G> {
121 /// Produces a proof of a instance given its satisfying witness `W`.
122 pub fn prove(W: &RelaxedR1CSWitness<G>) -> Result<FinalSNARK<G>, NovaError> {
123 Ok(FinalSNARK { W: W.clone() })
124 }
125
126 /// Verifies the proof of a folded instance `U` given its shape `S` public parameters `gens`
127 pub fn verify(
128 &self,
129 gens: &R1CSGens<G>,
130 S: &R1CSShape<G>,
131 U: &RelaxedR1CSInstance<G>,
132 ) -> Result<(), NovaError> {
133 // check that the witness is a valid witness to the folded instance `U`
134 S.is_sat_relaxed(gens, U, &self.W)
135 }
136}
137
138#[cfg(test)]
139mod tests {
140 use super::*;
141 use ::bellperson::{gadgets::num::AllocatedNum, ConstraintSystem, SynthesisError};
142 use ff::{Field, PrimeField};
143 use rand::rngs::OsRng;
144
145 type S = pasta_curves::pallas::Scalar;
146 type G = pasta_curves::pallas::Point;
147
148 fn synthesize_tiny_r1cs_bellperson<Scalar: PrimeField, CS: ConstraintSystem<Scalar>>(
149 cs: &mut CS,
150 x_val: Option<Scalar>,
151 ) -> Result<(), SynthesisError> {
152 // Consider a cubic equation: `x^3 + x + 5 = y`, where `x` and `y` are respectively the input and output.
153 let x = AllocatedNum::alloc(cs.namespace(|| "x"), || Ok(x_val.unwrap()))?;
154 let _ = x.inputize(cs.namespace(|| "x is input"));
155
156 let x_sq = x.square(cs.namespace(|| "x_sq"))?;
157 let x_cu = x_sq.mul(cs.namespace(|| "x_cu"), &x)?;
158 let y = AllocatedNum::alloc(cs.namespace(|| "y"), || {
159 Ok(x_cu.get_value().unwrap() + x.get_value().unwrap() + Scalar::from(5u64))
160 })?;
161 let _ = y.inputize(cs.namespace(|| "y is output"));
162
163 cs.enforce(
164 || "y = x^3 + x + 5",
165 |lc| {
166 lc + x_cu.get_variable()
167 + x.get_variable()
168 + CS::one()
169 + CS::one()
170 + CS::one()
171 + CS::one()
172 + CS::one()
173 },
174 |lc| lc + CS::one(),
175 |lc| lc + y.get_variable(),
176 );
177
178 Ok(())
179 }
180
181 #[test]
182 fn test_tiny_r1cs_bellperson() {
183 use super::bellperson::{
184 r1cs::{NovaShape, NovaWitness},
185 shape_cs::ShapeCS,
186 solver::SatisfyingAssignment,
187 };
188
189 // First create the shape
190 let mut cs: ShapeCS<G> = ShapeCS::new();
191 let _ = synthesize_tiny_r1cs_bellperson(&mut cs, None);
192 let shape = cs.r1cs_shape();
193 let gens = cs.r1cs_gens();
194
195 // Now get the instance and assignment for one instance
196 let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
197 let _ = synthesize_tiny_r1cs_bellperson(&mut cs, Some(S::from(5)));
198 let (U1, W1) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
199
200 // Make sure that the first instance is satisfiable
201 assert!(shape.is_sat(&gens, &U1, &W1).is_ok());
202
203 // Now get the instance and assignment for second instance
204 let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
205 let _ = synthesize_tiny_r1cs_bellperson(&mut cs, Some(S::from(135)));
206 let (U2, W2) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
207
208 // Make sure that the second instance is satisfiable
209 assert!(shape.is_sat(&gens, &U2, &W2).is_ok());
210
211 // execute a sequence of folds
212 execute_sequence(&gens, &shape, &U1, &W1, &U2, &W2);
213 }
214
215 fn execute_sequence(
216 gens: &R1CSGens<G>,
217 shape: &R1CSShape<G>,
218 U1: &R1CSInstance<G>,
219 W1: &R1CSWitness<G>,
220 U2: &R1CSInstance<G>,
221 W2: &R1CSWitness<G>,
222 ) {
223 // produce a default running instance
224 let mut r_W = RelaxedR1CSWitness::default(shape);
225 let mut r_U = RelaxedR1CSInstance::default(gens, shape);
226
227 // produce a step SNARK with (W1, U1) as the first incoming witness-instance pair
228 let mut prover_transcript = Transcript::new(b"StepSNARKExample");
229 let res = StepSNARK::prove(gens, shape, &r_U, &r_W, U1, W1, &mut prover_transcript);
230 assert!(res.is_ok());
231 let (step_snark, (_U, W)) = res.unwrap();
232
233 // verify the step SNARK with U1 as the first incoming instance
234 let mut verifier_transcript = Transcript::new(b"StepSNARKExample");
235 let res = step_snark.verify(&r_U, U1, &mut verifier_transcript);
236 assert!(res.is_ok());
237 let U = res.unwrap();
238
239 assert_eq!(U, _U);
240
241 // update the running witness and instance
242 r_W = W;
243 r_U = U;
244
245 // produce a step SNARK with (W2, U2) as the second incoming witness-instance pair
246 let res = StepSNARK::prove(gens, shape, &r_U, &r_W, U2, W2, &mut prover_transcript);
247 assert!(res.is_ok());
248 let (step_snark, (_U, W)) = res.unwrap();
249
250 // verify the step SNARK with U1 as the first incoming instance
251 let res = step_snark.verify(&r_U, U2, &mut verifier_transcript);
252 assert!(res.is_ok());
253 let U = res.unwrap();
254
255 assert_eq!(U, _U);
256
257 // update the running witness and instance
258 r_W = W;
259 r_U = U;
260
261 // produce a final SNARK
262 let res = FinalSNARK::prove(&r_W);
263 assert!(res.is_ok());
264 let final_snark = res.unwrap();
265 // verify the final SNARK
266 let res = final_snark.verify(gens, shape, &r_U);
267 assert!(res.is_ok());
268 }
269
270 #[test]
271 fn test_tiny_r1cs() {
272 let one = S::one();
273 let (num_cons, num_vars, num_io, A, B, C) = {
274 let num_cons = 4;
275 let num_vars = 4;
276 let num_io = 2;
277
278 // Consider a cubic equation: `x^3 + x + 5 = y`, where `x` and `y` are respectively the input and output.
279 // The R1CS for this problem consists of the following constraints:
280 // `I0 * I0 - Z0 = 0`
281 // `Z0 * I0 - Z1 = 0`
282 // `(Z1 + I0) * 1 - Z2 = 0`
283 // `(Z2 + 5) * 1 - I1 = 0`
284
285 // Relaxed R1CS is a set of three sparse matrices (A B C), where there is a row for every
286 // constraint and a column for every entry in z = (vars, u, inputs)
287 // An R1CS instance is satisfiable iff:
288 // Az \circ Bz = u \cdot Cz + E, where z = (vars, 1, inputs)
289 let mut A: Vec<(usize, usize, S)> = Vec::new();
290 let mut B: Vec<(usize, usize, S)> = Vec::new();
291 let mut C: Vec<(usize, usize, S)> = Vec::new();
292
293 // constraint 0 entries in (A,B,C)
294 // `I0 * I0 - Z0 = 0`
295 A.push((0, num_vars + 1, one));
296 B.push((0, num_vars + 1, one));
297 C.push((0, 0, one));
298
299 // constraint 1 entries in (A,B,C)
300 // `Z0 * I0 - Z1 = 0`
301 A.push((1, 0, one));
302 B.push((1, num_vars + 1, one));
303 C.push((1, 1, one));
304
305 // constraint 2 entries in (A,B,C)
306 // `(Z1 + I0) * 1 - Z2 = 0`
307 A.push((2, 1, one));
308 A.push((2, num_vars + 1, one));
309 B.push((2, num_vars, one));
310 C.push((2, 2, one));
311
312 // constraint 3 entries in (A,B,C)
313 // `(Z2 + 5) * 1 - I1 = 0`
314 A.push((3, 2, one));
315 A.push((3, num_vars, one + one + one + one + one));
316 B.push((3, num_vars, one));
317 C.push((3, num_vars + 2, one));
318
319 (num_cons, num_vars, num_io, A, B, C)
320 };
321
322 // create a shape object
323 let S = {
324 let res = R1CSShape::new(num_cons, num_vars, num_io, &A, &B, &C);
325 assert!(res.is_ok());
326 res.unwrap()
327 };
328
329 // generate generators
330 let gens = R1CSGens::new(num_cons, num_vars);
331
332 let rand_inst_witness_generator =
333 |gens: &R1CSGens<G>, I: &S| -> (S, R1CSInstance<G>, R1CSWitness<G>) {
334 let i0 = *I;
335
336 // compute a satisfying (vars, X) tuple
337 let (O, vars, X) = {
338 let z0 = i0 * i0; // constraint 0
339 let z1 = i0 * z0; // constraint 1
340 let z2 = z1 + i0; // constraint 2
341 let i1 = z2 + one + one + one + one + one; // constraint 3
342
343 // store the witness and IO for the instance
344 let W = vec![z0, z1, z2, S::zero()];
345 let X = vec![i0, i1];
346 (i1, W, X)
347 };
348
349 let W = {
350 let res = R1CSWitness::new(&S, &vars);
351 assert!(res.is_ok());
352 res.unwrap()
353 };
354 let U = {
355 let comm_W = W.commit(gens);
356 let res = R1CSInstance::new(&S, &comm_W, &X);
357 assert!(res.is_ok());
358 res.unwrap()
359 };
360
361 // check that generated instance is satisfiable
362 assert!(S.is_sat(gens, &U, &W).is_ok());
363
364 (O, U, W)
365 };
366
367 let mut csprng: OsRng = OsRng;
368 let I = S::random(&mut csprng); // the first input is picked randomly for the first instance
369 let (O, U1, W1) = rand_inst_witness_generator(&gens, &I);
370 let (_O, U2, W2) = rand_inst_witness_generator(&gens, &O);
371
372 // execute a sequence of folds
373 execute_sequence(&gens, &S, &U1, &W1, &U2, &W2);
374 }
375}