Nova/src/nifs.rs
src/nifs.rsRaw
1//! This module implements a non-interactive folding scheme
2#![allow(non_snake_case)]
3#![allow(clippy::type_complexity)]
4
5use super::commitments::CompressedCommitment;
6use super::errors::NovaError;
7use super::r1cs::{
8 R1CSGens, R1CSInstance, R1CSShape, R1CSWitness, RelaxedR1CSInstance, RelaxedR1CSWitness,
9};
10use super::traits::{AbsorbInROTrait, Group, HashFuncTrait};
11use std::marker::PhantomData;
12
13/// A SNARK that holds the proof of a step of an incremental computation
14#[allow(clippy::upper_case_acronyms)]
15pub struct NIFS<G: Group> {
16 pub(crate) comm_T: CompressedCommitment<G::CompressedGroupElement>,
17 _p: PhantomData<G>,
18}
19
20type HashFuncConstants<G> =
21 <<G as Group>::HashFunc as HashFuncTrait<<G as Group>::Base, <G as Group>::Scalar>>::Constants;
22
23impl<G: Group> NIFS<G> {
24 /// Takes as input a Relaxed R1CS instance-witness tuple `(U1, W1)` and
25 /// an R1CS instance-witness tuple `(U2, W2)` with the same structure `shape`
26 /// and defined with respect to the same `gens`, and outputs
27 /// a folded Relaxed R1CS instance-witness tuple `(U, W)` of the same shape `shape`,
28 /// with the guarantee that the folded witness `W` satisfies the folded instance `U`
29 /// if and only if `W1` satisfies `U1` and `W2` satisfies `U2`.
30 pub fn prove(
31 gens: &R1CSGens<G>,
32 ro_consts: &HashFuncConstants<G>,
33 S: &R1CSShape<G>,
34 U1: &RelaxedR1CSInstance<G>,
35 W1: &RelaxedR1CSWitness<G>,
36 U2: &R1CSInstance<G>,
37 W2: &R1CSWitness<G>,
38 ) -> Result<(NIFS<G>, (RelaxedR1CSInstance<G>, RelaxedR1CSWitness<G>)), NovaError> {
39 // initialize a new RO
40 let mut ro = G::HashFunc::new(ro_consts.clone());
41
42 // append S to the transcript
43 S.absorb_in_ro(&mut ro);
44
45 // append U1 and U2 to transcript
46 U1.absorb_in_ro(&mut ro);
47 U2.absorb_in_ro(&mut ro);
48
49 // compute a commitment to the cross-term
50 let (T, comm_T) = S.commit_T(gens, U1, W1, U2, W2)?;
51
52 // append `comm_T` to the transcript and obtain a challenge
53 comm_T.absorb_in_ro(&mut ro);
54
55 // compute a challenge from the RO
56 let r = ro.get_challenge();
57
58 // fold the instance using `r` and `comm_T`
59 let U = U1.fold(U2, &comm_T, &r)?;
60
61 // fold the witness using `r` and `T`
62 let W = W1.fold(W2, &T, &r)?;
63
64 // return the folded instance and witness
65 Ok((
66 Self {
67 comm_T: comm_T.compress(),
68 _p: Default::default(),
69 },
70 (U, W),
71 ))
72 }
73
74 /// Takes as input a relaxed R1CS instance `U1` and and R1CS instance `U2`
75 /// with the same shape and defined with respect to the same parameters,
76 /// and outputs a folded instance `U` with the same shape,
77 /// with the guarantee that the folded instance `U`
78 /// if and only if `U1` and `U2` are satisfiable.
79 pub fn verify(
80 &self,
81 ro_consts: &HashFuncConstants<G>,
82 S: &R1CSShape<G>,
83 U1: &RelaxedR1CSInstance<G>,
84 U2: &R1CSInstance<G>,
85 ) -> Result<RelaxedR1CSInstance<G>, NovaError> {
86 // initialize a new RO
87 let mut ro = G::HashFunc::new(ro_consts.clone());
88
89 // append S to the transcript
90 S.absorb_in_ro(&mut ro);
91
92 // append U1 and U2 to transcript
93 U1.absorb_in_ro(&mut ro);
94 U2.absorb_in_ro(&mut ro);
95
96 // append `comm_T` to the transcript and obtain a challenge
97 let comm_T = self.comm_T.decompress()?;
98 comm_T.absorb_in_ro(&mut ro);
99
100 // compute a challenge from the RO
101 let r = ro.get_challenge();
102
103 // fold the instance using `r` and `comm_T`
104 let U = U1.fold(U2, &comm_T, &r)?;
105
106 // return the folded instance
107 Ok(U)
108 }
109}
110
111#[cfg(test)]
112mod tests {
113 use super::*;
114 use crate::traits::{Group, HashFuncConstantsTrait};
115 use ::bellperson::{gadgets::num::AllocatedNum, ConstraintSystem, SynthesisError};
116 use ff::{Field, PrimeField};
117 use rand::rngs::OsRng;
118
119 type S = pasta_curves::pallas::Scalar;
120 type G = pasta_curves::pallas::Point;
121
122 fn synthesize_tiny_r1cs_bellperson<Scalar: PrimeField, CS: ConstraintSystem<Scalar>>(
123 cs: &mut CS,
124 x_val: Option<Scalar>,
125 ) -> Result<(), SynthesisError> {
126 // Consider a cubic equation: `x^3 + x + 5 = y`, where `x` and `y` are respectively the input and output.
127 let x = AllocatedNum::alloc(cs.namespace(|| "x"), || Ok(x_val.unwrap()))?;
128 let _ = x.inputize(cs.namespace(|| "x is input"));
129
130 let x_sq = x.square(cs.namespace(|| "x_sq"))?;
131 let x_cu = x_sq.mul(cs.namespace(|| "x_cu"), &x)?;
132 let y = AllocatedNum::alloc(cs.namespace(|| "y"), || {
133 Ok(x_cu.get_value().unwrap() + x.get_value().unwrap() + Scalar::from(5u64))
134 })?;
135 let _ = y.inputize(cs.namespace(|| "y is output"));
136
137 cs.enforce(
138 || "y = x^3 + x + 5",
139 |lc| {
140 lc + x_cu.get_variable()
141 + x.get_variable()
142 + CS::one()
143 + CS::one()
144 + CS::one()
145 + CS::one()
146 + CS::one()
147 },
148 |lc| lc + CS::one(),
149 |lc| lc + y.get_variable(),
150 );
151
152 Ok(())
153 }
154
155 #[test]
156 fn test_tiny_r1cs_bellperson() {
157 use crate::bellperson::{
158 r1cs::{NovaShape, NovaWitness},
159 shape_cs::ShapeCS,
160 solver::SatisfyingAssignment,
161 };
162
163 // First create the shape
164 let mut cs: ShapeCS<G> = ShapeCS::new();
165 let _ = synthesize_tiny_r1cs_bellperson(&mut cs, None);
166 let shape = cs.r1cs_shape();
167 let gens = cs.r1cs_gens();
168 let ro_consts = <<G as Group>::HashFunc as HashFuncTrait<
169 <G as Group>::Base,
170 <G as Group>::Scalar,
171 >>::Constants::new();
172
173 // Now get the instance and assignment for one instance
174 let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
175 let _ = synthesize_tiny_r1cs_bellperson(&mut cs, Some(S::from(5)));
176 let (U1, W1) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
177
178 // Make sure that the first instance is satisfiable
179 assert!(shape.is_sat(&gens, &U1, &W1).is_ok());
180
181 // Now get the instance and assignment for second instance
182 let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
183 let _ = synthesize_tiny_r1cs_bellperson(&mut cs, Some(S::from(135)));
184 let (U2, W2) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
185
186 // Make sure that the second instance is satisfiable
187 assert!(shape.is_sat(&gens, &U2, &W2).is_ok());
188
189 // execute a sequence of folds
190 execute_sequence(&gens, &ro_consts, &shape, &U1, &W1, &U2, &W2);
191 }
192
193 fn execute_sequence(
194 gens: &R1CSGens<G>,
195 ro_consts: &<<G as Group>::HashFunc as HashFuncTrait<
196 <G as Group>::Base,
197 <G as Group>::Scalar,
198 >>::Constants,
199 shape: &R1CSShape<G>,
200 U1: &R1CSInstance<G>,
201 W1: &R1CSWitness<G>,
202 U2: &R1CSInstance<G>,
203 W2: &R1CSWitness<G>,
204 ) {
205 // produce a default running instance
206 let mut r_W = RelaxedR1CSWitness::default(shape);
207 let mut r_U = RelaxedR1CSInstance::default(gens, shape);
208
209 // produce a step SNARK with (W1, U1) as the first incoming witness-instance pair
210 let res = NIFS::prove(gens, ro_consts, shape, &r_U, &r_W, U1, W1);
211 assert!(res.is_ok());
212 let (nifs, (_U, W)) = res.unwrap();
213
214 // verify the step SNARK with U1 as the first incoming instance
215 let res = nifs.verify(ro_consts, shape, &r_U, U1);
216 assert!(res.is_ok());
217 let U = res.unwrap();
218
219 assert_eq!(U, _U);
220
221 // update the running witness and instance
222 r_W = W;
223 r_U = U;
224
225 // produce a step SNARK with (W2, U2) as the second incoming witness-instance pair
226 let res = NIFS::prove(gens, ro_consts, shape, &r_U, &r_W, U2, W2);
227 assert!(res.is_ok());
228 let (nifs, (_U, W)) = res.unwrap();
229
230 // verify the step SNARK with U1 as the first incoming instance
231 let res = nifs.verify(ro_consts, shape, &r_U, U2);
232 assert!(res.is_ok());
233 let U = res.unwrap();
234
235 assert_eq!(U, _U);
236
237 // update the running witness and instance
238 r_W = W;
239 r_U = U;
240
241 // check if the running instance is satisfiable
242 assert!(shape.is_sat_relaxed(gens, &r_U, &r_W).is_ok());
243 }
244
245 #[test]
246 fn test_tiny_r1cs() {
247 let one = S::one();
248 let (num_cons, num_vars, num_io, A, B, C) = {
249 let num_cons = 4;
250 let num_vars = 4;
251 let num_io = 2;
252
253 // Consider a cubic equation: `x^3 + x + 5 = y`, where `x` and `y` are respectively the input and output.
254 // The R1CS for this problem consists of the following constraints:
255 // `I0 * I0 - Z0 = 0`
256 // `Z0 * I0 - Z1 = 0`
257 // `(Z1 + I0) * 1 - Z2 = 0`
258 // `(Z2 + 5) * 1 - I1 = 0`
259
260 // Relaxed R1CS is a set of three sparse matrices (A B C), where there is a row for every
261 // constraint and a column for every entry in z = (vars, u, inputs)
262 // An R1CS instance is satisfiable iff:
263 // Az \circ Bz = u \cdot Cz + E, where z = (vars, 1, inputs)
264 let mut A: Vec<(usize, usize, S)> = Vec::new();
265 let mut B: Vec<(usize, usize, S)> = Vec::new();
266 let mut C: Vec<(usize, usize, S)> = Vec::new();
267
268 // constraint 0 entries in (A,B,C)
269 // `I0 * I0 - Z0 = 0`
270 A.push((0, num_vars + 1, one));
271 B.push((0, num_vars + 1, one));
272 C.push((0, 0, one));
273
274 // constraint 1 entries in (A,B,C)
275 // `Z0 * I0 - Z1 = 0`
276 A.push((1, 0, one));
277 B.push((1, num_vars + 1, one));
278 C.push((1, 1, one));
279
280 // constraint 2 entries in (A,B,C)
281 // `(Z1 + I0) * 1 - Z2 = 0`
282 A.push((2, 1, one));
283 A.push((2, num_vars + 1, one));
284 B.push((2, num_vars, one));
285 C.push((2, 2, one));
286
287 // constraint 3 entries in (A,B,C)
288 // `(Z2 + 5) * 1 - I1 = 0`
289 A.push((3, 2, one));
290 A.push((3, num_vars, one + one + one + one + one));
291 B.push((3, num_vars, one));
292 C.push((3, num_vars + 2, one));
293
294 (num_cons, num_vars, num_io, A, B, C)
295 };
296
297 // create a shape object
298 let S = {
299 let res = R1CSShape::new(num_cons, num_vars, num_io, &A, &B, &C);
300 assert!(res.is_ok());
301 res.unwrap()
302 };
303
304 // generate generators and ro constants
305 let gens = R1CSGens::new(num_cons, num_vars);
306 let ro_consts = <<G as Group>::HashFunc as HashFuncTrait<
307 <G as Group>::Base,
308 <G as Group>::Scalar,
309 >>::Constants::new();
310
311 let rand_inst_witness_generator =
312 |gens: &R1CSGens<G>, I: &S| -> (S, R1CSInstance<G>, R1CSWitness<G>) {
313 let i0 = *I;
314
315 // compute a satisfying (vars, X) tuple
316 let (O, vars, X) = {
317 let z0 = i0 * i0; // constraint 0
318 let z1 = i0 * z0; // constraint 1
319 let z2 = z1 + i0; // constraint 2
320 let i1 = z2 + one + one + one + one + one; // constraint 3
321
322 // store the witness and IO for the instance
323 let W = vec![z0, z1, z2, S::zero()];
324 let X = vec![i0, i1];
325 (i1, W, X)
326 };
327
328 let W = {
329 let res = R1CSWitness::new(&S, &vars);
330 assert!(res.is_ok());
331 res.unwrap()
332 };
333 let U = {
334 let comm_W = W.commit(gens);
335 let res = R1CSInstance::new(&S, &comm_W, &X);
336 assert!(res.is_ok());
337 res.unwrap()
338 };
339
340 // check that generated instance is satisfiable
341 assert!(S.is_sat(gens, &U, &W).is_ok());
342
343 (O, U, W)
344 };
345
346 let mut csprng: OsRng = OsRng;
347 let I = S::random(&mut csprng); // the first input is picked randomly for the first instance
348 let (O, U1, W1) = rand_inst_witness_generator(&gens, &I);
349 let (_O, U2, W2) = rand_inst_witness_generator(&gens, &O);
350
351 // execute a sequence of folds
352 execute_sequence(&gens, &ro_consts, &S, &U1, &W1, &U2, &W2);
353 }
354}