Reorganize various Spartan SNARKs and make the direct interface more generic (#195)

* reorganize different variants of spartan and make direct snark more generic

* cargo fmt

src/spartan/snark.rs

@@ -0,0 +1,532 @@
+//! This module implements RelaxedR1CSSNARKTrait using Spartan that is generic
+//! over the polynomial commitment and evaluation argument (i.e., a PCS)
+//! This version of Spartan does not use preprocessing so the verifier keeps the entire
+//! description of R1CS matrices. This is essentially optimal for the verifier when using
+//! an IPA-based polynomial commitment scheme.
+
+use crate::{
+  compute_digest,
+  errors::NovaError,
+  r1cs::{R1CSShape, RelaxedR1CSInstance, RelaxedR1CSWitness},
+  spartan::{
+    polynomial::{EqPolynomial, MultilinearPolynomial, SparsePolynomial},
+    powers,
+    sumcheck::SumcheckProof,
+    PolyEvalInstance, PolyEvalWitness,
+  },
+  traits::{
+    evaluation::EvaluationEngineTrait, snark::RelaxedR1CSSNARKTrait, Group, TranscriptEngineTrait,
+  },
+  Commitment, CommitmentKey,
+};
+use ff::Field;
+use itertools::concat;
+use rayon::prelude::*;
+use serde::{Deserialize, Serialize};
+
+/// A type that represents the prover's key
+#[derive(Serialize, Deserialize)]
+#[serde(bound = "")]
+pub struct ProverKey<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
+  pk_ee: EE::ProverKey,
+  S: R1CSShape<G>,
+  vk_digest: G::Scalar, // digest of the verifier's key
+}
+
+/// A type that represents the verifier's key
+#[derive(Serialize, Deserialize)]
+#[serde(bound = "")]
+pub struct VerifierKey<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
+  vk_ee: EE::VerifierKey,
+  S: R1CSShape<G>,
+  digest: G::Scalar,
+}
+
+/// A succinct proof of knowledge of a witness to a relaxed R1CS instance
+/// The proof is produced using Spartan's combination of the sum-check and
+/// the commitment to a vector viewed as a polynomial commitment
+#[derive(Serialize, Deserialize)]
+#[serde(bound = "")]
+pub struct RelaxedR1CSSNARK<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
+  sc_proof_outer: SumcheckProof<G>,
+  claims_outer: (G::Scalar, G::Scalar, G::Scalar),
+  eval_E: G::Scalar,
+  sc_proof_inner: SumcheckProof<G>,
+  eval_W: G::Scalar,
+  sc_proof_batch: SumcheckProof<G>,
+  evals_batch: Vec<G::Scalar>,
+  eval_arg: EE::EvaluationArgument,
+}
+
+impl<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> RelaxedR1CSSNARKTrait<G>
+  for RelaxedR1CSSNARK<G, EE>
+{
+  type ProverKey = ProverKey<G, EE>;
+  type VerifierKey = VerifierKey<G, EE>;
+
+  fn setup(
+    ck: &CommitmentKey<G>,
+    S: &R1CSShape<G>,
+  ) -> Result<(Self::ProverKey, Self::VerifierKey), NovaError> {
+    let (pk_ee, vk_ee) = EE::setup(ck);
+
+    let S = S.pad();
+
+    let vk = {
+      let mut vk = VerifierKey {
+        vk_ee,
+        S: S.clone(),
+        digest: G::Scalar::ZERO,
+      };
+      vk.digest = compute_digest::<G, VerifierKey<G, EE>>(&vk);
+      vk
+    };
+
+    let pk = ProverKey {
+      pk_ee,
+      S,
+      vk_digest: vk.digest,
+    };
+
+    Ok((pk, vk))
+  }
+
+  /// produces a succinct proof of satisfiability of a RelaxedR1CS instance
+  fn prove(
+    ck: &CommitmentKey<G>,
+    pk: &Self::ProverKey,
+    U: &RelaxedR1CSInstance<G>,
+    W: &RelaxedR1CSWitness<G>,
+  ) -> Result<Self, NovaError> {
+    let W = W.pad(&pk.S); // pad the witness
+    let mut transcript = G::TE::new(b"RelaxedR1CSSNARK");
+
+    // sanity check that R1CSShape has certain size characteristics
+    assert_eq!(pk.S.num_cons.next_power_of_two(), pk.S.num_cons);
+    assert_eq!(pk.S.num_vars.next_power_of_two(), pk.S.num_vars);
+    assert_eq!(pk.S.num_io.next_power_of_two(), pk.S.num_io);
+    assert!(pk.S.num_io < pk.S.num_vars);
+
+    // append the digest of vk (which includes R1CS matrices) and the RelaxedR1CSInstance to the transcript
+    transcript.absorb(b"vk", &pk.vk_digest);
+    transcript.absorb(b"U", U);
+
+    // compute the full satisfying assignment by concatenating W.W, U.u, and U.X
+    let mut z = concat(vec![W.W.clone(), vec![U.u], U.X.clone()]);
+
+    let (num_rounds_x, num_rounds_y) = (
+      (pk.S.num_cons as f64).log2() as usize,
+      ((pk.S.num_vars as f64).log2() as usize + 1),
+    );
+
+    // outer sum-check
+    let tau = (0..num_rounds_x)
+      .map(|_i| transcript.squeeze(b"t"))
+      .collect::<Result<Vec<G::Scalar>, NovaError>>()?;
+
+    let mut poly_tau = MultilinearPolynomial::new(EqPolynomial::new(tau).evals());
+    let (mut poly_Az, mut poly_Bz, poly_Cz, mut poly_uCz_E) = {
+      let (poly_Az, poly_Bz, poly_Cz) = pk.S.multiply_vec(&z)?;
+      let poly_uCz_E = (0..pk.S.num_cons)
+        .map(|i| U.u * poly_Cz[i] + W.E[i])
+        .collect::<Vec<G::Scalar>>();
+      (
+        MultilinearPolynomial::new(poly_Az),
+        MultilinearPolynomial::new(poly_Bz),
+        MultilinearPolynomial::new(poly_Cz),
+        MultilinearPolynomial::new(poly_uCz_E),
+      )
+    };
+
+    let comb_func_outer =
+      |poly_A_comp: &G::Scalar,
+       poly_B_comp: &G::Scalar,
+       poly_C_comp: &G::Scalar,
+       poly_D_comp: &G::Scalar|
+       -> G::Scalar { *poly_A_comp * (*poly_B_comp * *poly_C_comp - *poly_D_comp) };
+    let (sc_proof_outer, r_x, claims_outer) = SumcheckProof::prove_cubic_with_additive_term(
+      &G::Scalar::ZERO, // claim is zero
+      num_rounds_x,
+      &mut poly_tau,
+      &mut poly_Az,
+      &mut poly_Bz,
+      &mut poly_uCz_E,
+      comb_func_outer,
+      &mut transcript,
+    )?;
+
+    // claims from the end of sum-check
+    let (claim_Az, claim_Bz): (G::Scalar, G::Scalar) = (claims_outer[1], claims_outer[2]);
+    let claim_Cz = poly_Cz.evaluate(&r_x);
+    let eval_E = MultilinearPolynomial::new(W.E.clone()).evaluate(&r_x);
+    transcript.absorb(
+      b"claims_outer",
+      &[claim_Az, claim_Bz, claim_Cz, eval_E].as_slice(),
+    );
+
+    // inner sum-check
+    let r = transcript.squeeze(b"r")?;
+    let claim_inner_joint = claim_Az + r * claim_Bz + r * r * claim_Cz;
+
+    let poly_ABC = {
+      // compute the initial evaluation table for R(\tau, x)
+      let evals_rx = EqPolynomial::new(r_x.clone()).evals();
+
+      // Bounds "row" variables of (A, B, C) matrices viewed as 2d multilinear polynomials
+      let compute_eval_table_sparse =
+        |S: &R1CSShape<G>, rx: &[G::Scalar]| -> (Vec<G::Scalar>, Vec<G::Scalar>, Vec<G::Scalar>) {
+          assert_eq!(rx.len(), S.num_cons);
+
+          let inner = |M: &Vec<(usize, usize, G::Scalar)>, M_evals: &mut Vec<G::Scalar>| {
+            for (row, col, val) in M {
+              M_evals[*col] += rx[*row] * val;
+            }
+          };
+
+          let (A_evals, (B_evals, C_evals)) = rayon::join(
+            || {
+              let mut A_evals: Vec<G::Scalar> = vec![G::Scalar::ZERO; 2 * S.num_vars];
+              inner(&S.A, &mut A_evals);
+              A_evals
+            },
+            || {
+              rayon::join(
+                || {
+                  let mut B_evals: Vec<G::Scalar> = vec![G::Scalar::ZERO; 2 * S.num_vars];
+                  inner(&S.B, &mut B_evals);
+                  B_evals
+                },
+                || {
+                  let mut C_evals: Vec<G::Scalar> = vec![G::Scalar::ZERO; 2 * S.num_vars];
+                  inner(&S.C, &mut C_evals);
+                  C_evals
+                },
+              )
+            },
+          );
+
+          (A_evals, B_evals, C_evals)
+        };
+
+      let (evals_A, evals_B, evals_C) = compute_eval_table_sparse(&pk.S, &evals_rx);
+
+      assert_eq!(evals_A.len(), evals_B.len());
+      assert_eq!(evals_A.len(), evals_C.len());
+      (0..evals_A.len())
+        .into_par_iter()
+        .map(|i| evals_A[i] + r * evals_B[i] + r * r * evals_C[i])
+        .collect::<Vec<G::Scalar>>()
+    };
+
+    let poly_z = {
+      z.resize(pk.S.num_vars * 2, G::Scalar::ZERO);
+      z
+    };
+
+    let comb_func = |poly_A_comp: &G::Scalar, poly_B_comp: &G::Scalar| -> G::Scalar {
+      *poly_A_comp * *poly_B_comp
+    };
+    let (sc_proof_inner, r_y, _claims_inner) = SumcheckProof::prove_quad(
+      &claim_inner_joint,
+      num_rounds_y,
+      &mut MultilinearPolynomial::new(poly_ABC),
+      &mut MultilinearPolynomial::new(poly_z),
+      comb_func,
+      &mut transcript,
+    )?;
+
+    // add additional claims about W and E polynomials to the list from CC
+    let mut w_u_vec = Vec::new();
+    let eval_W = MultilinearPolynomial::evaluate_with(&W.W, &r_y[1..]);
+    w_u_vec.push((
+      PolyEvalWitness { p: W.W.clone() },
+      PolyEvalInstance {
+        c: U.comm_W,
+        x: r_y[1..].to_vec(),
+        e: eval_W,
+      },
+    ));
+
+    w_u_vec.push((
+      PolyEvalWitness { p: W.E },
+      PolyEvalInstance {
+        c: U.comm_E,
+        x: r_x,
+        e: eval_E,
+      },
+    ));
+
+    // We will now reduce a vector of claims of evaluations at different points into claims about them at the same point.
+    // For example, eval_W =? W(r_y[1..]) and eval_E =? E(r_x) into
+    // two claims: eval_W_prime =? W(rz) and eval_E_prime =? E(rz)
+    // We can them combine the two into one: eval_W_prime + gamma * eval_E_prime =? (W + gamma*E)(rz),
+    // where gamma is a public challenge
+    // Since commitments to W and E are homomorphic, the verifier can compute a commitment
+    // to the batched polynomial.
+    assert!(w_u_vec.len() >= 2);
+
+    let (w_vec, u_vec): (Vec<PolyEvalWitness<G>>, Vec<PolyEvalInstance<G>>) =
+      w_u_vec.into_iter().unzip();
+    let w_vec_padded = PolyEvalWitness::pad(&w_vec); // pad the polynomials to be of the same size
+    let u_vec_padded = PolyEvalInstance::pad(&u_vec); // pad the evaluation points
+
+    // generate a challenge
+    let rho = transcript.squeeze(b"r")?;
+    let num_claims = w_vec_padded.len();
+    let powers_of_rho = powers::<G>(&rho, num_claims);
+    let claim_batch_joint = u_vec_padded
+      .iter()
+      .zip(powers_of_rho.iter())
+      .map(|(u, p)| u.e * p)
+      .sum();
+
+    let mut polys_left: Vec<MultilinearPolynomial<G::Scalar>> = w_vec_padded
+      .iter()
+      .map(|w| MultilinearPolynomial::new(w.p.clone()))
+      .collect();
+    let mut polys_right: Vec<MultilinearPolynomial<G::Scalar>> = u_vec_padded
+      .iter()
+      .map(|u| MultilinearPolynomial::new(EqPolynomial::new(u.x.clone()).evals()))
+      .collect();
+
+    let num_rounds_z = u_vec_padded[0].x.len();
+    let comb_func = |poly_A_comp: &G::Scalar, poly_B_comp: &G::Scalar| -> G::Scalar {
+      *poly_A_comp * *poly_B_comp
+    };
+    let (sc_proof_batch, r_z, claims_batch) = SumcheckProof::prove_quad_batch(
+      &claim_batch_joint,
+      num_rounds_z,
+      &mut polys_left,
+      &mut polys_right,
+      &powers_of_rho,
+      comb_func,
+      &mut transcript,
+    )?;
+
+    let (claims_batch_left, _): (Vec<G::Scalar>, Vec<G::Scalar>) = claims_batch;
+
+    transcript.absorb(b"l", &claims_batch_left.as_slice());
+
+    // we now combine evaluation claims at the same point rz into one
+    let gamma = transcript.squeeze(b"g")?;
+    let powers_of_gamma: Vec<G::Scalar> = powers::<G>(&gamma, num_claims);
+    let comm_joint = u_vec_padded
+      .iter()
+      .zip(powers_of_gamma.iter())
+      .map(|(u, g_i)| u.c * *g_i)
+      .fold(Commitment::<G>::default(), |acc, item| acc + item);
+    let poly_joint = PolyEvalWitness::weighted_sum(&w_vec_padded, &powers_of_gamma);
+    let eval_joint = claims_batch_left
+      .iter()
+      .zip(powers_of_gamma.iter())
+      .map(|(e, g_i)| *e * *g_i)
+      .sum();
+
+    let eval_arg = EE::prove(
+      ck,
+      &pk.pk_ee,
+      &mut transcript,
+      &comm_joint,
+      &poly_joint.p,
+      &r_z,
+      &eval_joint,
+    )?;
+
+    Ok(RelaxedR1CSSNARK {
+      sc_proof_outer,
+      claims_outer: (claim_Az, claim_Bz, claim_Cz),
+      eval_E,
+      sc_proof_inner,
+      eval_W,
+      sc_proof_batch,
+      evals_batch: claims_batch_left,
+      eval_arg,
+    })
+  }
+
+  /// verifies a proof of satisfiability of a RelaxedR1CS instance
+  fn verify(&self, vk: &Self::VerifierKey, U: &RelaxedR1CSInstance<G>) -> Result<(), NovaError> {
+    let mut transcript = G::TE::new(b"RelaxedR1CSSNARK");
+
+    // append the digest of R1CS matrices and the RelaxedR1CSInstance to the transcript
+    transcript.absorb(b"vk", &vk.digest);
+    transcript.absorb(b"U", U);
+
+    let (num_rounds_x, num_rounds_y) = (
+      (vk.S.num_cons as f64).log2() as usize,
+      ((vk.S.num_vars as f64).log2() as usize + 1),
+    );
+
+    // outer sum-check
+    let tau = (0..num_rounds_x)
+      .map(|_i| transcript.squeeze(b"t"))
+      .collect::<Result<Vec<G::Scalar>, NovaError>>()?;
+
+    let (claim_outer_final, r_x) =
+      self
+        .sc_proof_outer
+        .verify(G::Scalar::ZERO, num_rounds_x, 3, &mut transcript)?;
+
+    // verify claim_outer_final
+    let (claim_Az, claim_Bz, claim_Cz) = self.claims_outer;
+    let taus_bound_rx = EqPolynomial::new(tau).evaluate(&r_x);
+    let claim_outer_final_expected =
+      taus_bound_rx * (claim_Az * claim_Bz - U.u * claim_Cz - self.eval_E);
+    if claim_outer_final != claim_outer_final_expected {
+      return Err(NovaError::InvalidSumcheckProof);
+    }
+
+    transcript.absorb(
+      b"claims_outer",
+      &[
+        self.claims_outer.0,
+        self.claims_outer.1,
+        self.claims_outer.2,
+        self.eval_E,
+      ]
+      .as_slice(),
+    );
+
+    // inner sum-check
+    let r = transcript.squeeze(b"r")?;
+    let claim_inner_joint =
+      self.claims_outer.0 + r * self.claims_outer.1 + r * r * self.claims_outer.2;
+
+    let (claim_inner_final, r_y) =
+      self
+        .sc_proof_inner
+        .verify(claim_inner_joint, num_rounds_y, 2, &mut transcript)?;
+
+    // verify claim_inner_final
+    let eval_Z = {
+      let eval_X = {
+        // constant term
+        let mut poly_X = vec![(0, U.u)];
+        //remaining inputs
+        poly_X.extend(
+          (0..U.X.len())
+            .map(|i| (i + 1, U.X[i]))
+            .collect::<Vec<(usize, G::Scalar)>>(),
+        );
+        SparsePolynomial::new((vk.S.num_vars as f64).log2() as usize, poly_X).evaluate(&r_y[1..])
+      };
+      (G::Scalar::ONE - r_y[0]) * self.eval_W + r_y[0] * eval_X
+    };
+
+    // compute evaluations of R1CS matrices
+    let multi_evaluate = |M_vec: &[&[(usize, usize, G::Scalar)]],
+                          r_x: &[G::Scalar],
+                          r_y: &[G::Scalar]|
+     -> Vec<G::Scalar> {
+      let evaluate_with_table =
+        |M: &[(usize, usize, G::Scalar)], T_x: &[G::Scalar], T_y: &[G::Scalar]| -> G::Scalar {
+          (0..M.len())
+            .collect::<Vec<usize>>()
+            .par_iter()
+            .map(|&i| {
+              let (row, col, val) = M[i];
+              T_x[row] * T_y[col] * val
+            })
+            .reduce(|| G::Scalar::ZERO, |acc, x| acc + x)
+        };
+
+      let (T_x, T_y) = rayon::join(
+        || EqPolynomial::new(r_x.to_vec()).evals(),
+        || EqPolynomial::new(r_y.to_vec()).evals(),
+      );
+
+      (0..M_vec.len())
+        .collect::<Vec<usize>>()
+        .par_iter()
+        .map(|&i| evaluate_with_table(M_vec[i], &T_x, &T_y))
+        .collect()
+    };
+
+    let evals = multi_evaluate(&[&vk.S.A, &vk.S.B, &vk.S.C], &r_x, &r_y);
+
+    let claim_inner_final_expected = (evals[0] + r * evals[1] + r * r * evals[2]) * eval_Z;
+    if claim_inner_final != claim_inner_final_expected {
+      return Err(NovaError::InvalidSumcheckProof);
+    }
+
+    // add claims about W and E polynomials
+    let u_vec: Vec<PolyEvalInstance<G>> = vec![
+      PolyEvalInstance {
+        c: U.comm_W,
+        x: r_y[1..].to_vec(),
+        e: self.eval_W,
+      },
+      PolyEvalInstance {
+        c: U.comm_E,
+        x: r_x,
+        e: self.eval_E,
+      },
+    ];
+
+    let u_vec_padded = PolyEvalInstance::pad(&u_vec); // pad the evaluation points
+
+    // generate a challenge
+    let rho = transcript.squeeze(b"r")?;
+    let num_claims = u_vec.len();
+    let powers_of_rho = powers::<G>(&rho, num_claims);
+    let claim_batch_joint = u_vec
+      .iter()
+      .zip(powers_of_rho.iter())
+      .map(|(u, p)| u.e * p)
+      .sum();
+
+    let num_rounds_z = u_vec_padded[0].x.len();
+    let (claim_batch_final, r_z) =
+      self
+        .sc_proof_batch
+        .verify(claim_batch_joint, num_rounds_z, 2, &mut transcript)?;
+
+    let claim_batch_final_expected = {
+      let poly_rz = EqPolynomial::new(r_z.clone());
+      let evals = u_vec_padded
+        .iter()
+        .map(|u| poly_rz.evaluate(&u.x))
+        .collect::<Vec<G::Scalar>>();
+
+      evals
+        .iter()
+        .zip(self.evals_batch.iter())
+        .zip(powers_of_rho.iter())
+        .map(|((e_i, p_i), rho_i)| *e_i * *p_i * rho_i)
+        .sum()
+    };
+
+    if claim_batch_final != claim_batch_final_expected {
+      return Err(NovaError::InvalidSumcheckProof);
+    }
+
+    transcript.absorb(b"l", &self.evals_batch.as_slice());
+
+    // we now combine evaluation claims at the same point rz into one
+    let gamma = transcript.squeeze(b"g")?;
+    let powers_of_gamma: Vec<G::Scalar> = powers::<G>(&gamma, num_claims);
+    let comm_joint = u_vec_padded
+      .iter()
+      .zip(powers_of_gamma.iter())
+      .map(|(u, g_i)| u.c * *g_i)
+      .fold(Commitment::<G>::default(), |acc, item| acc + item);
+    let eval_joint = self
+      .evals_batch
+      .iter()
+      .zip(powers_of_gamma.iter())
+      .map(|(e, g_i)| *e * *g_i)
+      .sum();
+
+    // verify
+    EE::verify(
+      &vk.vk_ee,
+      &mut transcript,
+      &comm_joint,
+      &r_z,
+      &eval_joint,
+      &self.eval_arg,
+    )?;
+
+    Ok(())
+  }
+}