initial commit

src/bullet.rs

@@ -0,0 +1,247 @@
+#![allow(non_snake_case)]
+
+use super::errors::ProofVerifyError;
+use super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
+use super::math::Math;
+use super::scalar::Scalar;
+use super::transcript::ProofTranscript;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+use std::iter;
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct BulletReductionProof {
+  L_vec: Vec<CompressedGroup>,
+  R_vec: Vec<CompressedGroup>,
+}
+
+impl BulletReductionProof {
+  /// Create an inner-product proof.
+  ///
+  /// The proof is created with respect to the bases \\(G\\).
+  ///
+  /// The `transcript` is passed in as a parameter so that the
+  /// challenges depend on the *entire* transcript (including parent
+  /// protocols).
+  ///
+  /// The lengths of the vectors must all be the same, and must all be
+  /// either 0 or a power of 2.
+  pub fn prove(
+    transcript: &mut Transcript,
+    Q: &GroupElement,
+    G_vec: &Vec<GroupElement>,
+    H: &GroupElement,
+    a_vec: &Vec<Scalar>,
+    b_vec: &Vec<Scalar>,
+    blind: &Scalar,
+    blinds_vec: &Vec<(Scalar, Scalar)>,
+  ) -> (
+    BulletReductionProof,
+    GroupElement,
+    Scalar,
+    Scalar,
+    GroupElement,
+    Scalar,
+  ) {
+    // Create slices G, H, a, b backed by their respective
+    // vectors.  This lets us reslice as we compress the lengths
+    // of the vectors in the main loop below.
+    let mut G = &mut G_vec.clone()[..];
+    let mut a = &mut a_vec.clone()[..];
+    let mut b = &mut b_vec.clone()[..];
+
+    // All of the input vectors must have a length that is a power of two.
+    let mut n = G.len();
+    assert!(n.is_power_of_two());
+    let lg_n = n.log2();
+
+    let G_factors: Vec<Scalar> = iter::repeat(Scalar::one()).take(n).collect();
+
+    // All of the input vectors must have the same length.
+    assert_eq!(G.len(), n);
+    assert_eq!(a.len(), n);
+    assert_eq!(b.len(), n);
+    assert_eq!(G_factors.len(), n);
+    assert_eq!(blinds_vec.len(), 2 * lg_n);
+
+    //transcript.innerproduct_domain_sep(n as u64);
+
+    let mut L_vec = Vec::with_capacity(lg_n);
+    let mut R_vec = Vec::with_capacity(lg_n);
+    let mut blinds_iter = blinds_vec.iter();
+    let mut blind_fin = *blind;
+
+    while n != 1 {
+      n = n / 2;
+      let (a_L, a_R) = a.split_at_mut(n);
+      let (b_L, b_R) = b.split_at_mut(n);
+      let (G_L, G_R) = G.split_at_mut(n);
+
+      let c_L = inner_product(&a_L, &b_R);
+      let c_R = inner_product(&a_R, &b_L);
+
+      let (blind_L, blind_R) = blinds_iter.next().unwrap();
+
+      let L = GroupElement::vartime_multiscalar_mul(
+        a_L
+          .iter()
+          .chain(iter::once(&c_L))
+          .chain(iter::once(blind_L)),
+        G_R.iter().chain(iter::once(Q)).chain(iter::once(H)),
+      );
+
+      let R = GroupElement::vartime_multiscalar_mul(
+        a_R
+          .iter()
+          .chain(iter::once(&c_R))
+          .chain(iter::once(blind_R)),
+        G_L.iter().chain(iter::once(Q)).chain(iter::once(H)),
+      );
+
+      transcript.append_point(b"L", &L.compress());
+      transcript.append_point(b"R", &R.compress());
+
+      let u = transcript.challenge_scalar(b"u");
+      let u_inv = u.invert().unwrap();
+
+      for i in 0..n {
+        a_L[i] = a_L[i] * u + u_inv * a_R[i];
+        b_L[i] = b_L[i] * u_inv + u * b_R[i];
+        G_L[i] = GroupElement::vartime_multiscalar_mul(&[u_inv, u], &[G_L[i], G_R[i]]);
+      }
+
+      blind_fin = blind_fin + blind_L * &u * &u + blind_R * &u_inv * &u_inv;
+
+      L_vec.push(L.compress());
+      R_vec.push(R.compress());
+
+      a = a_L;
+      b = b_L;
+      G = G_L;
+    }
+
+    let Gamma_hat =
+      GroupElement::vartime_multiscalar_mul(&[a[0], a[0] * b[0], blind_fin], &[G[0], *Q, *H]);
+
+    (
+      BulletReductionProof {
+        L_vec: L_vec,
+        R_vec: R_vec,
+      },
+      Gamma_hat,
+      a[0],
+      b[0],
+      G[0],
+      blind_fin,
+    )
+  }
+
+  /// Computes three vectors of verification scalars \\([u\_{i}^{2}]\\), \\([u\_{i}^{-2}]\\) and \\([s\_{i}]\\) for combined multiscalar multiplication
+  /// in a parent protocol. See [inner product protocol notes](index.html#verification-equation) for details.
+  /// The verifier must provide the input length \\(n\\) explicitly to avoid unbounded allocation within the inner product proof.
+  fn verification_scalars(
+    &self,
+    n: usize,
+    transcript: &mut Transcript,
+  ) -> Result<(Vec<Scalar>, Vec<Scalar>, Vec<Scalar>), ProofVerifyError> {
+    let lg_n = self.L_vec.len();
+    if lg_n >= 32 {
+      // 4 billion multiplications should be enough for anyone
+      // and this check prevents overflow in 1<<lg_n below.
+      return Err(ProofVerifyError);
+    }
+    if n != (1 << lg_n) {
+      return Err(ProofVerifyError);
+    }
+
+    // 1. Recompute x_k,...,x_1 based on the proof transcript
+    let mut challenges = Vec::with_capacity(lg_n);
+    for (L, R) in self.L_vec.iter().zip(self.R_vec.iter()) {
+      transcript.append_point(b"L", L);
+      transcript.append_point(b"R", R);
+      challenges.push(transcript.challenge_scalar(b"u"));
+    }
+
+    // 2. Compute 1/(u_k...u_1) and 1/u_k, ..., 1/u_1
+    let mut challenges_inv = challenges.clone();
+    let allinv = Scalar::batch_invert(&mut challenges_inv);
+
+    // 3. Compute u_i^2 and (1/u_i)^2
+    for i in 0..lg_n {
+      challenges[i] = challenges[i] * challenges[i];
+      challenges_inv[i] = challenges_inv[i] * challenges_inv[i];
+    }
+    let challenges_sq = challenges;
+    let challenges_inv_sq = challenges_inv;
+
+    // 4. Compute s values inductively.
+    let mut s = Vec::with_capacity(n);
+    s.push(allinv);
+    for i in 1..n {
+      let lg_i = (32 - 1 - (i as u32).leading_zeros()) as usize;
+      let k = 1 << lg_i;
+      // The challenges are stored in "creation order" as [u_k,...,u_1],
+      // so u_{lg(i)+1} = is indexed by (lg_n-1) - lg_i
+      let u_lg_i_sq = challenges_sq[(lg_n - 1) - lg_i];
+      s.push(s[i - k] * u_lg_i_sq);
+    }
+
+    Ok((challenges_sq, challenges_inv_sq, s))
+  }
+
+  /// This method is for testing that proof generation work,
+  /// but for efficiency the actual protocols would use `verification_scalars`
+  /// method to combine inner product verification with other checks
+  /// in a single multiscalar multiplication.
+  pub fn verify(
+    &self,
+    n: usize,
+    a: &Vec<Scalar>,
+    transcript: &mut Transcript,
+    Gamma: &GroupElement,
+    G: &[GroupElement],
+  ) -> Result<(GroupElement, GroupElement, Scalar), ProofVerifyError> {
+    let (u_sq, u_inv_sq, s) = self.verification_scalars(n, transcript)?;
+
+    let Ls = self
+      .L_vec
+      .iter()
+      .map(|p| p.decompress().ok_or(ProofVerifyError))
+      .collect::<Result<Vec<_>, _>>()?;
+
+    let Rs = self
+      .R_vec
+      .iter()
+      .map(|p| p.decompress().ok_or(ProofVerifyError))
+      .collect::<Result<Vec<_>, _>>()?;
+
+    let G_hat = GroupElement::vartime_multiscalar_mul(s.iter(), G.iter());
+    let a_hat = inner_product(a, &s);
+
+    let Gamma_hat = GroupElement::vartime_multiscalar_mul(
+      u_sq
+        .iter()
+        .chain(u_inv_sq.iter())
+        .chain(iter::once(&Scalar::one())),
+      Ls.iter().chain(Rs.iter()).chain(iter::once(Gamma)),
+    );
+
+    Ok((G_hat, Gamma_hat, a_hat))
+  }
+}
+
+/// Computes an inner product of two vectors
+/// \\[
+///    {\langle {\mathbf{a}}, {\mathbf{b}} \rangle} = \sum\_{i=0}^{n-1} a\_i \cdot b\_i.
+/// \\]
+/// Panics if the lengths of \\(\mathbf{a}\\) and \\(\mathbf{b}\\) are not equal.
+pub fn inner_product(a: &[Scalar], b: &[Scalar]) -> Scalar {
+  let mut out = Scalar::zero();
+  if a.len() != b.len() {
+    panic!("inner_product(a,b): lengths of vectors do not match");
+  }
+  for i in 0..a.len() {
+    out += a[i] * b[i];
+  }
+  out
+}

src/commitments.rs

@@ -0,0 +1,109 @@
+use super::group::{GroupElement, VartimeMultiscalarMul, GROUP_BASEPOINT_COMPRESSED};
+use super::scalar::Scalar;
+use digest::{ExtendableOutput, Input, XofReader};
+use sha3::Shake256;
+
+#[derive(Debug)]
+pub struct MultiCommitGens {
+  pub n: usize,
+  pub G: Vec<GroupElement>,
+  pub h: GroupElement,
+}
+
+impl MultiCommitGens {
+  pub fn new(n: usize, label: &[u8]) -> Self {
+    let mut shake = Shake256::default();
+    shake.input(label);
+    shake.input(GROUP_BASEPOINT_COMPRESSED.as_bytes());
+
+    let mut reader = shake.xof_result();
+    let mut gens: Vec<GroupElement> = Vec::new();
+    let mut uniform_bytes = [0u8; 64];
+    for _ in 0..n + 1 {
+      reader.read(&mut uniform_bytes);
+      gens.push(GroupElement::from_uniform_bytes(&uniform_bytes));
+    }
+
+    MultiCommitGens {
+      n,
+      G: gens[0..n].to_vec(),
+      h: gens[n],
+    }
+  }
+
+  pub fn clone(&self) -> MultiCommitGens {
+    MultiCommitGens {
+      n: self.n,
+      h: self.h,
+      G: self.G.clone(),
+    }
+  }
+
+  pub fn split_at_mut(&mut self, mid: usize) -> (MultiCommitGens, MultiCommitGens) {
+    let (G1, G2) = self.G.split_at_mut(mid);
+
+    (
+      MultiCommitGens {
+        n: G1.len(),
+        G: G1.to_vec(),
+        h: self.h,
+      },
+      MultiCommitGens {
+        n: G2.len(),
+        G: G2.to_vec(),
+        h: self.h,
+      },
+    )
+  }
+}
+
+pub trait Commitments {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement;
+}
+
+impl Commitments for Scalar {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement {
+    assert!(gens_n.n == 1);
+    GroupElement::vartime_multiscalar_mul(&[*self, *blind], &[gens_n.G[0], gens_n.h])
+  }
+}
+
+impl Commitments for Vec<Scalar> {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement {
+    assert!(gens_n.n == self.len());
+    GroupElement::vartime_multiscalar_mul(self, &gens_n.G) + blind * &gens_n.h
+  }
+}
+
+impl Commitments for [Scalar] {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement {
+    assert_eq!(gens_n.n, self.len());
+    GroupElement::vartime_multiscalar_mul(self, &gens_n.G) + blind * &gens_n.h
+  }
+}
+
+impl Commitments for Vec<bool> {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement {
+    assert!(gens_n.n == self.len());
+    let mut comm = blind * &gens_n.h;
+    for i in 0..self.len() {
+      if self[i] {
+        comm = comm + gens_n.G[i];
+      }
+    }
+    comm
+  }
+}
+
+impl Commitments for [bool] {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement {
+    assert!(gens_n.n == self.len());
+    let mut comm = blind * &gens_n.h;
+    for i in 0..self.len() {
+      if self[i] {
+        comm = comm + gens_n.G[i];
+      }
+    }
+    comm
+  }
+}

src/dense_mlpoly.rs

@@ -0,0 +1,740 @@
+use super::commitments::{Commitments, MultiCommitGens};
+use super::errors::ProofVerifyError;
+use super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
+use super::math::Math;
+use super::nizk::{DotProductProofGens, DotProductProofLog};
+use super::scalar::Scalar;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use core::ops::Index;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+#[cfg(feature = "rayon_par")]
+use rayon::prelude::*;
+
+#[derive(Debug)]
+pub struct DensePolynomial {
+  num_vars: usize, //the number of variables in the multilinear polynomial
+  len: usize,
+  Z: Vec<Scalar>, // a vector that holds the evaluations of the polynomial in all the 2^num_vars Boolean inputs
+}
+
+pub struct PolyCommitmentGens {
+  pub gens: DotProductProofGens,
+}
+
+impl PolyCommitmentGens {
+  // the number of variables in the multilinear polynomial
+  pub fn new(num_vars: usize, label: &'static [u8]) -> PolyCommitmentGens {
+    let (_left, right) = EqPolynomial::compute_factored_lens(num_vars);
+    let gens = DotProductProofGens::new(right.pow2(), label);
+    PolyCommitmentGens { gens }
+  }
+}
+
+pub struct PolyCommitmentBlinds {
+  blinds: Vec<Scalar>,
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct PolyCommitment {
+  C: Vec<CompressedGroup>,
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct ConstPolyCommitment {
+  C: CompressedGroup,
+}
+
+impl PolyCommitment {
+  pub fn combine(&self, comm: &PolyCommitment, s: &Scalar) -> PolyCommitment {
+    assert_eq!(comm.C.len(), self.C.len());
+    let C = (0..self.C.len())
+      .map(|i| (self.C[i].decompress().unwrap() + s * comm.C[i].decompress().unwrap()).compress())
+      .collect::<Vec<CompressedGroup>>();
+    PolyCommitment { C }
+  }
+
+  pub fn combine_const(&self, comm: &ConstPolyCommitment) -> PolyCommitment {
+    let C = (0..self.C.len())
+      .map(|i| (self.C[i].decompress().unwrap() + comm.C.decompress().unwrap()).compress())
+      .collect::<Vec<CompressedGroup>>();
+    PolyCommitment { C }
+  }
+}
+
+pub struct EqPolynomial {
+  r: Vec<Scalar>,
+}
+
+impl EqPolynomial {
+  pub fn new(r: Vec<Scalar>) -> Self {
+    EqPolynomial { r }
+  }
+
+  pub fn evaluate(&self, rx: &Vec<Scalar>) -> Scalar {
+    assert_eq!(self.r.len(), rx.len());
+    (0..rx.len())
+      .map(|i| self.r[i] * rx[i] + (Scalar::one() - self.r[i]) * (Scalar::one() - rx[i]))
+      .product()
+  }
+
+  pub fn evals(&self) -> Vec<Scalar> {
+    let ell = self.r.len();
+
+    let mut evals: Vec<Scalar> = vec![Scalar::one(); ell.pow2()];
+    let mut size = 1;
+    for j in 0..ell {
+      // in each iteration, we double the size of chis
+      size = size * 2;
+      for i in (0..size).rev().step_by(2) {
+        // copy each element from the prior iteration twice
+        let scalar = evals[i / 2];
+        //                evals[i - 1] = scalar * (Scalar::one() - tau[j]);
+        //                evals[i] = scalar * tau[j];
+        evals[i] = scalar * self.r[j];
+        evals[i - 1] = scalar - evals[i];
+      }
+    }
+    evals
+  }
+
+  pub fn compute_factored_lens(ell: usize) -> (usize, usize) {
+    (ell / 2, ell - ell / 2)
+  }
+
+  pub fn compute_factored_evals(&self) -> (Vec<Scalar>, Vec<Scalar>) {
+    let ell = self.r.len();
+    let (left_num_vars, _right_num_vars) = EqPolynomial::compute_factored_lens(ell);
+
+    let L = EqPolynomial::new(self.r[0..left_num_vars].to_vec()).evals();
+    let R = EqPolynomial::new(self.r[left_num_vars..ell].to_vec()).evals();
+
+    (L, R)
+  }
+}
+
+pub struct ConstPolynomial {
+  num_vars: usize,
+  c: Scalar,
+}
+
+impl ConstPolynomial {
+  pub fn new(num_vars: usize, c: Scalar) -> Self {
+    ConstPolynomial { num_vars, c }
+  }
+
+  pub fn evaluate(&self, rx: &Vec<Scalar>) -> Scalar {
+    assert_eq!(self.num_vars, rx.len());
+    self.c
+  }
+
+  pub fn get_num_vars(&self) -> usize {
+    self.num_vars
+  }
+
+  /// produces a binding commitment
+  pub fn commit(&self, gens: &PolyCommitmentGens) -> PolyCommitment {
+    let ell = self.get_num_vars();
+
+    let (left_num_vars, right_num_vars) = EqPolynomial::compute_factored_lens(ell);
+    let L_size = left_num_vars.pow2();
+    let R_size = right_num_vars.pow2();
+    assert_eq!(L_size * R_size, ell.pow2());
+
+    let vec = vec![self.c; R_size];
+
+    let c = vec.commit(&Scalar::zero(), &gens.gens.gens_n).compress();
+    PolyCommitment { C: vec![c; L_size] }
+  }
+}
+
+pub struct IdentityPolynomial {
+  size_point: usize,
+}
+
+impl IdentityPolynomial {
+  pub fn new(size_point: usize) -> Self {
+    IdentityPolynomial { size_point }
+  }
+
+  pub fn evaluate(&self, r: &Vec<Scalar>) -> Scalar {
+    let len = r.len();
+    assert_eq!(len, self.size_point);
+    (0..len)
+      .map(|i| Scalar::from((len - i - 1).pow2() as u64) * r[i])
+      .sum()
+  }
+}
+
+impl DensePolynomial {
+  pub fn new(Z: Vec<Scalar>) -> Self {
+    let len = Z.len();
+    let num_vars = len.log2();
+    DensePolynomial { num_vars, Z, len }
+  }
+
+  pub fn get_num_vars(&self) -> usize {
+    self.num_vars
+  }
+
+  pub fn len(&self) -> usize {
+    self.len
+  }
+
+  pub fn clone(&self) -> DensePolynomial {
+    DensePolynomial::new(self.Z[0..self.len].to_vec())
+  }
+
+  pub fn split(&self, idx: usize) -> (DensePolynomial, DensePolynomial) {
+    assert!(idx < self.len());
+    (
+      DensePolynomial::new(self.Z[0..idx].to_vec()),
+      DensePolynomial::new(self.Z[idx..2 * idx].to_vec()),
+    )
+  }
+
+  #[cfg(feature = "rayon_par")]
+  fn commit_inner(&self, blinds: &Vec<Scalar>, gens: &MultiCommitGens) -> PolyCommitment {
+    let L_size = blinds.len();
+    let R_size = self.Z.len() / L_size;
+    assert_eq!(L_size * R_size, self.Z.len());
+    let C = (0..L_size)
+      .collect::<Vec<usize>>()
+      .par_iter()
+      .map(|&i| {
+        self.Z[R_size * i..R_size * (i + 1)]
+          .commit(&blinds[i], gens)
+          .compress()
+      })
+      .collect();
+    PolyCommitment { C }
+  }
+
+  #[cfg(not(feature = "rayon_par"))]
+  fn commit_inner(&self, blinds: &Vec<Scalar>, gens: &MultiCommitGens) -> PolyCommitment {
+    let L_size = blinds.len();
+    let R_size = self.Z.len() / L_size;
+    assert_eq!(L_size * R_size, self.Z.len());
+    let C = (0..L_size)
+      .map(|i| {
+        self.Z[R_size * i..R_size * (i + 1)]
+          .commit(&blinds[i], gens)
+          .compress()
+      })
+      .collect();
+    PolyCommitment { C }
+  }
+
+  pub fn commit(
+    &self,
+    hiding: bool,
+    gens: &PolyCommitmentGens,
+    random_tape: Option<&mut Transcript>,
+  ) -> (PolyCommitment, PolyCommitmentBlinds) {
+    let n = self.Z.len();
+    let ell = self.get_num_vars();
+    assert_eq!(n, ell.pow2());
+
+    let (left_num_vars, right_num_vars) = EqPolynomial::compute_factored_lens(ell);
+    let L_size = left_num_vars.pow2();
+    let R_size = right_num_vars.pow2();
+    assert_eq!(L_size * R_size, n);
+
+    let blinds = match hiding {
+      true => PolyCommitmentBlinds {
+        blinds: random_tape
+          .unwrap()
+          .challenge_vector(b"poly_blinds", L_size),
+      },
+      false => PolyCommitmentBlinds {
+        blinds: vec![Scalar::zero(); L_size],
+      },
+    };
+
+    (self.commit_inner(&blinds.blinds, &gens.gens.gens_n), blinds)
+  }
+
+  pub fn bound(&self, L: &Vec<Scalar>) -> Vec<Scalar> {
+    let (left_num_vars, right_num_vars) = EqPolynomial::compute_factored_lens(self.get_num_vars());
+    let L_size = left_num_vars.pow2();
+    let R_size = right_num_vars.pow2();
+    (0..R_size)
+      .map(|i| (0..L_size).map(|j| &L[j] * &self.Z[j * R_size + i]).sum())
+      .collect::<Vec<Scalar>>()
+  }
+
+  pub fn bound_poly_var_top(&mut self, r: &Scalar) {
+    let n = self.len() / 2;
+    for i in 0..n {
+      self.Z[i] = &self.Z[i] + r * (&self.Z[i + n] - &self.Z[i]);
+    }
+    self.num_vars = self.num_vars - 1;
+    self.len = n;
+  }
+
+  pub fn bound_poly_var_bot(&mut self, r: &Scalar) {
+    let n = self.len() / 2;
+    for i in 0..n {
+      self.Z[i] = &self.Z[2 * i] + r * (&self.Z[2 * i + 1] - &self.Z[2 * i]);
+    }
+    self.num_vars = self.num_vars - 1;
+    self.len = n;
+  }
+
+  pub fn dotproduct(&self, other: &DensePolynomial) -> Scalar {
+    assert_eq!(self.len(), other.len());
+    let mut res = Scalar::zero();
+    for i in 0..self.len() {
+      res = &res + &self.Z[i] * &other[i];
+    }
+    res
+  }
+
+  // returns Z(r) in O(n) time
+  pub fn evaluate(&self, r: &Vec<Scalar>) -> Scalar {
+    // r must have a value for each variable
+    assert_eq!(r.len(), self.get_num_vars());
+    let chis = EqPolynomial::new(r.to_vec()).evals();
+    assert_eq!(chis.len(), self.Z.len());
+    DotProductProofLog::compute_dotproduct(&self.Z, &chis)
+  }
+
+  fn vec(&self) -> &Vec<Scalar> {
+    &self.Z
+  }
+
+  pub fn extend(&mut self, other: &DensePolynomial) {
+    // TODO: allow extension even when some vars are bound
+    assert_eq!(self.Z.len(), self.len);
+    let other_vec = other.vec();
+    assert_eq!(other_vec.len(), self.len);
+    self.Z.extend(other_vec);
+    self.num_vars = self.num_vars + 1;
+    self.len = 2 * self.len;
+    assert_eq!(self.Z.len(), self.len);
+  }
+
+  pub fn merge<'a, I>(polys: I) -> DensePolynomial
+  where
+    I: IntoIterator<Item = &'a DensePolynomial>,
+  {
+    //assert!(polys.len() > 0);
+    //let num_vars = polys[0].num_vars();
+    let mut Z: Vec<Scalar> = Vec::new();
+    for poly in polys.into_iter() {
+      //assert_eq!(poly.get_num_vars(), num_vars); // ensure each polynomial has the same number of variables
+      //assert_eq!(poly.len, poly.vec().len()); // ensure no variable is already bound
+      Z.extend(poly.vec());
+    }
+
+    // pad the polynomial with zero polynomial at the end
+    Z.resize(Z.len().next_power_of_two(), Scalar::zero());
+
+    DensePolynomial::new(Z)
+  }
+
+  pub fn from_usize(Z: &Vec<usize>) -> Self {
+    DensePolynomial::new(
+      (0..Z.len())
+        .map(|i| Scalar::from(Z[i] as u64))
+        .collect::<Vec<Scalar>>(),
+    )
+  }
+}
+
+impl Index<usize> for DensePolynomial {
+  type Output = Scalar;
+
+  #[inline(always)]
+  fn index(&self, _index: usize) -> &Scalar {
+    &(self.Z[_index])
+  }
+}
+
+impl AppendToTranscript for PolyCommitment {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_message(label, b"poly_commitment_begin");
+    for i in 0..self.C.len() {
+      transcript.append_point(b"poly_commitment_share", &self.C[i]);
+    }
+    transcript.append_message(label, b"poly_commitment_end");
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct PolyEvalProof {
+  proof: DotProductProofLog,
+}
+
+impl PolyEvalProof {
+  fn protocol_name() -> &'static [u8] {
+    b"polynomial evaluation proof"
+  }
+
+  pub fn prove(
+    poly: &DensePolynomial,
+    blinds_opt: Option<&PolyCommitmentBlinds>,
+    r: &Vec<Scalar>,               // point at which the polynomial is evaluated
+    Zr: &Scalar,                   // evaluation of \widetilde{Z}(r)
+    blind_Zr_opt: Option<&Scalar>, // specifies a blind for Zr
+    gens: &PolyCommitmentGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+  ) -> (PolyEvalProof, CompressedGroup) {
+    transcript.append_protocol_name(PolyEvalProof::protocol_name());
+
+    // assert vectors are of the right size
+    assert_eq!(poly.get_num_vars(), r.len());
+
+    let (left_num_vars, right_num_vars) = EqPolynomial::compute_factored_lens(r.len());
+    let L_size = left_num_vars.pow2();
+    let R_size = right_num_vars.pow2();
+
+    let default_blinds = PolyCommitmentBlinds {
+      blinds: vec![Scalar::zero(); L_size],
+    };
+    let blinds = match blinds_opt {
+      Some(p) => p,
+      None => &default_blinds,
+    };
+
+    assert_eq!(blinds.blinds.len(), L_size);
+
+    let zero = Scalar::zero();
+    let blind_Zr = match blind_Zr_opt {
+      Some(p) => p,
+      None => &zero,
+    };
+
+    // compute the L and R vectors
+    let eq = EqPolynomial::new(r.to_vec());
+    let (L, R) = eq.compute_factored_evals();
+    assert_eq!(L.len(), L_size);
+    assert_eq!(R.len(), R_size);
+
+    // compute the vector underneath L*Z and the L*blinds
+    // compute vector-matrix product between L and Z viewed as a matrix
+    let LZ = poly.bound(&L);
+    let LZ_blind: Scalar = (0..L.len()).map(|i| blinds.blinds[i] * L[i]).sum();
+
+    // a dot product proof of size R_size
+    let (proof, _C_LR, C_Zr_prime) = DotProductProofLog::prove(
+      &gens.gens,
+      transcript,
+      random_tape,
+      &LZ,
+      &LZ_blind,
+      &R,
+      &Zr,
+      blind_Zr,
+    );
+
+    (PolyEvalProof { proof }, C_Zr_prime)
+  }
+
+  pub fn verify(
+    &self,
+    gens: &PolyCommitmentGens,
+    transcript: &mut Transcript,
+    r: &Vec<Scalar>,        // point at which the polynomial is evaluated
+    C_Zr: &CompressedGroup, // commitment to \widetilde{Z}(r)
+    comm: &PolyCommitment,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(PolyEvalProof::protocol_name());
+
+    // compute L and R
+    let eq = EqPolynomial::new(r.to_vec());
+    let (L, R) = eq.compute_factored_evals();
+
+    // compute a weighted sum of commitments and L
+    let C_decompressed = comm.C.iter().map(|pt| pt.decompress().unwrap());
+
+    let C_LZ = GroupElement::vartime_multiscalar_mul(&L, C_decompressed).compress();
+
+    self
+      .proof
+      .verify(R.len(), &gens.gens, transcript, &R, &C_LZ, C_Zr)
+  }
+
+  pub fn verify_batched(
+    &self,
+    gens: &PolyCommitmentGens,
+    transcript: &mut Transcript,
+    r: &Vec<Scalar>,        // point at which the polynomial is evaluated
+    C_Zr: &CompressedGroup, // commitment to \widetilde{Z}(r)
+    comm: &[&PolyCommitment],
+    coeff: &[&Scalar],
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(PolyEvalProof::protocol_name());
+
+    // compute L and R
+    let eq = EqPolynomial::new(r.to_vec());
+    let (L, R) = eq.compute_factored_evals();
+
+    // compute a weighted sum of commitments and L
+    let C_decompressed: Vec<Vec<GroupElement>> = (0..comm.len())
+      .map(|i| {
+        comm[i]
+          .C
+          .iter()
+          .map(|pt| pt.decompress().unwrap())
+          .collect()
+      })
+      .collect();
+
+    let C_LZ: Vec<GroupElement> = (0..comm.len())
+      .map(|i| GroupElement::vartime_multiscalar_mul(&L, &C_decompressed[i]))
+      .collect();
+
+    let C_LZ_combined: GroupElement = (0..C_LZ.len()).map(|i| C_LZ[i] * coeff[i]).sum();
+
+    self.proof.verify(
+      R.len(),
+      &gens.gens,
+      transcript,
+      &R,
+      &C_LZ_combined.compress(),
+      C_Zr,
+    )
+  }
+
+  pub fn verify_plain(
+    &self,
+    gens: &PolyCommitmentGens,
+    transcript: &mut Transcript,
+    r: &Vec<Scalar>, // point at which the polynomial is evaluated
+    Zr: &Scalar,     // evaluation \widetilde{Z}(r)
+    comm: &PolyCommitment,
+  ) -> Result<(), ProofVerifyError> {
+    // compute a commitment to Zr with a blind of zero
+    let C_Zr = Zr.commit(&Scalar::zero(), &gens.gens.gens_1).compress();
+
+    self.verify(gens, transcript, r, &C_Zr, comm)
+  }
+
+  pub fn verify_plain_batched(
+    &self,
+    gens: &PolyCommitmentGens,
+    transcript: &mut Transcript,
+    r: &Vec<Scalar>, // point at which the polynomial is evaluated
+    Zr: &Scalar,     // evaluation \widetilde{Z}(r)
+    comm: &[&PolyCommitment],
+    coeff: &[&Scalar],
+  ) -> Result<(), ProofVerifyError> {
+    // compute a commitment to Zr with a blind of zero
+    let C_Zr = Zr.commit(&Scalar::zero(), &gens.gens.gens_1).compress();
+
+    assert_eq!(comm.len(), coeff.len());
+
+    self.verify_batched(gens, transcript, r, &C_Zr, comm, coeff)
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::super::scalar::ScalarFromPrimitives;
+  use super::*;
+  use rand::rngs::OsRng;
+
+  fn evaluate_with_LR(Z: &Vec<Scalar>, r: &Vec<Scalar>) -> Scalar {
+    let eq = EqPolynomial::new(r.to_vec());
+    let (L, R) = eq.compute_factored_evals();
+
+    let ell = r.len();
+    // ensure ell is even
+    assert!(ell % 2 == 0);
+    // compute n = 2^\ell
+    let n = ell.pow2();
+    // compute m = sqrt(n) = 2^{\ell/2}
+    let m = n.square_root();
+
+    // compute vector-matrix product between L and Z viewed as a matrix
+    let LZ = (0..m)
+      .map(|i| (0..m).map(|j| L[j] * Z[j * m + i]).sum())
+      .collect::<Vec<Scalar>>();
+
+    // compute dot product between LZ and R
+    DotProductProofLog::compute_dotproduct(&LZ, &R)
+  }
+
+  #[test]
+  fn check_polynomial_evaluation() {
+    let mut Z: Vec<Scalar> = Vec::new(); // Z = [1, 2, 1, 4]
+    Z.push(Scalar::one());
+    Z.push((2 as usize).to_scalar());
+    Z.push((1 as usize).to_scalar());
+    Z.push((4 as usize).to_scalar());
+    // r = [4,3]
+    let mut r: Vec<Scalar> = Vec::new();
+    r.push((4 as usize).to_scalar());
+    r.push((3 as usize).to_scalar());
+
+    let eval_with_LR = evaluate_with_LR(&Z, &r);
+    let poly = DensePolynomial::new(Z);
+
+    let eval = poly.evaluate(&r);
+    assert_eq!(eval, (28 as usize).to_scalar());
+    assert_eq!(eval_with_LR, eval);
+  }
+
+  pub fn compute_factored_chis_at_r(r: &Vec<Scalar>) -> (Vec<Scalar>, Vec<Scalar>) {
+    let mut L: Vec<Scalar> = Vec::new();
+    let mut R: Vec<Scalar> = Vec::new();
+
+    let ell = r.len();
+    assert!(ell % 2 == 0); // ensure ell is even
+    let n = ell.pow2();
+    let m = n.square_root();
+
+    // compute row vector L
+    for i in 0..m {
+      let mut chi_i = Scalar::one();
+      for j in 0..ell / 2 {
+        let bit_j = ((m * i) & (1 << (r.len() - j - 1))) > 0;
+        if bit_j {
+          chi_i *= r[j];
+        } else {
+          chi_i *= Scalar::one() - r[j];
+        }
+      }
+      L.push(chi_i);
+    }
+
+    // compute column vector R
+    for i in 0..m {
+      let mut chi_i = Scalar::one();
+      for j in ell / 2..ell {
+        let bit_j = (i & (1 << (r.len() - j - 1))) > 0;
+        if bit_j {
+          chi_i *= r[j];
+        } else {
+          chi_i *= Scalar::one() - r[j];
+        }
+      }
+      R.push(chi_i);
+    }
+    (L, R)
+  }
+
+  pub fn compute_chis_at_r(r: &Vec<Scalar>) -> Vec<Scalar> {
+    let ell = r.len();
+    let n = ell.pow2();
+    let mut chis: Vec<Scalar> = Vec::new();
+    for i in 0..n {
+      let mut chi_i = Scalar::one();
+      for j in 0..r.len() {
+        let bit_j = (i & (1 << (r.len() - j - 1))) > 0;
+        if bit_j {
+          chi_i *= r[j];
+        } else {
+          chi_i *= Scalar::one() - r[j];
+        }
+      }
+      chis.push(chi_i);
+    }
+    chis
+  }
+
+  pub fn compute_outerproduct(L: Vec<Scalar>, R: Vec<Scalar>) -> Vec<Scalar> {
+    assert_eq!(L.len(), R.len());
+
+    let mut O: Vec<Scalar> = Vec::new();
+    let m = L.len();
+    for i in 0..m {
+      for j in 0..m {
+        O.push(L[i] * R[j]);
+      }
+    }
+    O
+  }
+
+  #[test]
+  fn check_memoized_chis() {
+    let mut csprng: OsRng = OsRng;
+
+    let s = 10;
+    let mut r: Vec<Scalar> = Vec::new();
+    for _i in 0..s {
+      r.push(Scalar::random(&mut csprng));
+    }
+    let chis = tests::compute_chis_at_r(&r);
+    let chis_m = EqPolynomial::new(r).evals();
+    assert_eq!(chis, chis_m);
+  }
+
+  #[test]
+  fn check_factored_chis() {
+    let mut csprng: OsRng = OsRng;
+
+    let s = 10;
+    let mut r: Vec<Scalar> = Vec::new();
+    for _i in 0..s {
+      r.push(Scalar::random(&mut csprng));
+    }
+    let chis = EqPolynomial::new(r.clone()).evals();
+    let (L, R) = EqPolynomial::new(r).compute_factored_evals();
+    let O = compute_outerproduct(L, R);
+    assert_eq!(chis, O);
+  }
+
+  #[test]
+  fn check_memoized_factored_chis() {
+    let mut csprng: OsRng = OsRng;
+
+    let s = 10;
+    let mut r: Vec<Scalar> = Vec::new();
+    for _i in 0..s {
+      r.push(Scalar::random(&mut csprng));
+    }
+    let (L, R) = tests::compute_factored_chis_at_r(&r);
+    let eq = EqPolynomial::new(r);
+    let (L2, R2) = eq.compute_factored_evals();
+    assert_eq!(L, L2);
+    assert_eq!(R, R2);
+  }
+
+  #[test]
+  fn check_polynomial_commit() {
+    let mut Z: Vec<Scalar> = Vec::new(); // Z = [1, 2, 1, 4]
+    Z.push((1 as usize).to_scalar());
+    Z.push((2 as usize).to_scalar());
+    Z.push((1 as usize).to_scalar());
+    Z.push((4 as usize).to_scalar());
+
+    let poly = DensePolynomial::new(Z);
+
+    // r = [4,3]
+    let mut r: Vec<Scalar> = Vec::new();
+    r.push((4 as usize).to_scalar());
+    r.push((3 as usize).to_scalar());
+    let eval = poly.evaluate(&r);
+    assert_eq!(eval, (28 as usize).to_scalar());
+
+    let gens = PolyCommitmentGens::new(poly.get_num_vars(), b"test-two");
+    let (poly_commitment, blinds) = poly.commit(false, &gens, None);
+
+    let mut random_tape = {
+      let mut csprng: OsRng = OsRng;
+      let mut tape = Transcript::new(b"proof");
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut csprng));
+      tape
+    };
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, C_Zr) = PolyEvalProof::prove(
+      &poly,
+      Some(&blinds),
+      &r,
+      &eval,
+      None,
+      &gens,
+      &mut prover_transcript,
+      &mut random_tape,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens, &mut verifier_transcript, &r, &C_Zr, &poly_commitment)
+      .is_ok());
+  }
+}

src/errors.rs

@@ -0,0 +1,15 @@
+use std::fmt;
+
+pub struct ProofVerifyError;
+
+impl fmt::Display for ProofVerifyError {
+  fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+    write!(f, "Proof verification failed")
+  }
+}
+
+impl fmt::Debug for ProofVerifyError {
+  fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+    write!(f, "{{ file: {}, line: {} }}", file!(), line!())
+  }
+}

src/group.rs

@@ -0,0 +1,101 @@
+use super::scalar::{Scalar, ScalarBytes, ScalarBytesFromScalar};
+use core::borrow::Borrow;
+use core::ops::{Mul, MulAssign};
+
+pub type GroupElement = curve25519_dalek::ristretto::RistrettoPoint;
+pub type CompressedGroup = curve25519_dalek::ristretto::CompressedRistretto;
+pub const GROUP_BASEPOINT_COMPRESSED: CompressedGroup =
+  curve25519_dalek::constants::RISTRETTO_BASEPOINT_COMPRESSED;
+
+impl<'b> MulAssign<&'b Scalar> for GroupElement {
+  fn mul_assign(&mut self, scalar: &'b Scalar) {
+    let result = (self as &GroupElement) * Scalar::decompress_scalar(scalar);
+    *self = result;
+  }
+}
+
+impl<'a, 'b> Mul<&'b Scalar> for &'a GroupElement {
+  type Output = GroupElement;
+  fn mul(self, scalar: &'b Scalar) -> GroupElement {
+    self * Scalar::decompress_scalar(scalar)
+  }
+}
+
+impl<'a, 'b> Mul<&'b GroupElement> for &'a Scalar {
+  type Output = GroupElement;
+
+  fn mul(self, point: &'b GroupElement) -> GroupElement {
+    Scalar::decompress_scalar(self) * point
+  }
+}
+
+macro_rules! define_mul_variants {
+  (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {
+    impl<'b> Mul<&'b $rhs> for $lhs {
+      type Output = $out;
+      fn mul(self, rhs: &'b $rhs) -> $out {
+        &self * rhs
+      }
+    }
+
+    impl<'a> Mul<$rhs> for &'a $lhs {
+      type Output = $out;
+      fn mul(self, rhs: $rhs) -> $out {
+        self * &rhs
+      }
+    }
+
+    impl Mul<$rhs> for $lhs {
+      type Output = $out;
+      fn mul(self, rhs: $rhs) -> $out {
+        &self * &rhs
+      }
+    }
+  };
+}
+
+macro_rules! define_mul_assign_variants {
+  (LHS = $lhs:ty, RHS = $rhs:ty) => {
+    impl MulAssign<$rhs> for $lhs {
+      fn mul_assign(&mut self, rhs: $rhs) {
+        *self *= &rhs;
+      }
+    }
+  };
+}
+
+define_mul_assign_variants!(LHS = GroupElement, RHS = Scalar);
+define_mul_variants!(LHS = GroupElement, RHS = Scalar, Output = GroupElement);
+define_mul_variants!(LHS = Scalar, RHS = GroupElement, Output = GroupElement);
+
+pub trait VartimeMultiscalarMul {
+  type Scalar;
+  fn vartime_multiscalar_mul<I, J>(scalars: I, points: J) -> Self
+  where
+    I: IntoIterator,
+    I::Item: Borrow<Self::Scalar>,
+    J: IntoIterator,
+    J::Item: Borrow<Self>,
+    Self: Clone;
+}
+
+impl VartimeMultiscalarMul for GroupElement {
+  type Scalar = super::scalar::Scalar;
+  fn vartime_multiscalar_mul<I, J>(scalars: I, points: J) -> Self
+  where
+    I: IntoIterator,
+    I::Item: Borrow<Self::Scalar>,
+    J: IntoIterator,
+    J::Item: Borrow<Self>,
+    Self: Clone,
+  {
+    use curve25519_dalek::traits::VartimeMultiscalarMul;
+    <Self as VartimeMultiscalarMul>::vartime_multiscalar_mul(
+      scalars
+        .into_iter()
+        .map(|s| Scalar::decompress_scalar(s.borrow()))
+        .collect::<Vec<ScalarBytes>>(),
+      points,
+    )
+  }
+}

src/lib.rs

@@ -0,0 +1,31 @@
+#![allow(non_snake_case)]
+#![feature(test)]
+
+extern crate byteorder;
+extern crate core;
+extern crate curve25519_dalek;
+extern crate digest;
+extern crate merlin;
+extern crate rand;
+extern crate rayon;
+extern crate sha3;
+extern crate test;
+
+mod bullet;
+pub mod commitments;
+pub mod dense_mlpoly;
+mod errors;
+mod group;
+pub mod math;
+pub mod nizk;
+mod product_tree;
+pub mod r1csinstance;
+pub mod r1csproof;
+pub mod scalar;
+mod scalar_25519;
+pub mod sparse_mlpoly;
+pub mod spartan;
+pub mod sumcheck;
+pub mod timer;
+pub mod transcript;
+mod unipoly;

src/math.rs

@@ -0,0 +1,31 @@
+pub trait Math {
+  fn square_root(self) -> usize;
+  fn pow2(self) -> usize;
+  fn log2(self) -> usize;
+  fn get_bits(self, num_bits: usize) -> Vec<bool>;
+}
+
+impl Math for usize {
+  #[inline]
+  fn square_root(self) -> usize {
+    (self as f64).sqrt() as usize
+  }
+
+  #[inline]
+  fn pow2(self) -> usize {
+    let base: usize = 2;
+    base.pow(self as u32)
+  }
+
+  #[inline]
+  fn log2(self) -> usize {
+    (self as f64).log2() as usize
+  }
+
+  /// Returns the num_bits from n in a canonical order
+  fn get_bits(self, num_bits: usize) -> Vec<bool> {
+    (0..num_bits)
+      .map(|shift_amount| ((self & (1 << (num_bits - shift_amount - 1))) > 0))
+      .collect::<Vec<bool>>()
+  }
+}

src/nizk.rs

@@ -0,0 +1,750 @@
+use super::bullet::BulletReductionProof;
+use super::commitments::{Commitments, MultiCommitGens};
+use super::errors::ProofVerifyError;
+use super::group::CompressedGroup;
+use super::math::Math;
+use super::scalar::Scalar;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct KnowledgeProof {
+  alpha: CompressedGroup,
+  z1: Scalar,
+  z2: Scalar,
+}
+
+impl KnowledgeProof {
+  fn protocol_name() -> &'static [u8] {
+    b"knowledge proof"
+  }
+
+  pub fn prove(
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+    x: &Scalar,
+    r: &Scalar,
+  ) -> (KnowledgeProof, CompressedGroup) {
+    transcript.append_protocol_name(KnowledgeProof::protocol_name());
+
+    // produce two random Scalars
+    let t1 = random_tape.challenge_scalar(b"t1");
+    let t2 = random_tape.challenge_scalar(b"t2");
+
+    let C = x.commit(&r, gens_n).compress();
+    C.append_to_transcript(b"C", transcript);
+
+    let alpha = t1.commit(&t2, gens_n).compress();
+    alpha.append_to_transcript(b"alpha", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z1 = x * &c + &t1;
+    let z2 = r * &c + &t2;
+
+    (KnowledgeProof { alpha, z1, z2 }, C)
+  }
+
+  pub fn verify(
+    &self,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    C: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(KnowledgeProof::protocol_name());
+    C.append_to_transcript(b"C", transcript);
+    self.alpha.append_to_transcript(b"alpha", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let lhs = self.z1.commit(&self.z2, gens_n).compress();
+    let rhs = (&c * C.decompress().expect("Could not decompress C")
+      + self
+        .alpha
+        .decompress()
+        .expect("Could not decompress self.alpha"))
+    .compress();
+
+    if lhs == rhs {
+      Ok(())
+    } else {
+      Err(ProofVerifyError)
+    }
+  }
+}
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct EqualityProof {
+  alpha: CompressedGroup,
+  z: Scalar,
+}
+
+impl EqualityProof {
+  fn protocol_name() -> &'static [u8] {
+    b"equality proof"
+  }
+
+  pub fn prove(
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+    v1: &Scalar,
+    s1: &Scalar,
+    v2: &Scalar,
+    s2: &Scalar,
+  ) -> (EqualityProof, CompressedGroup, CompressedGroup) {
+    transcript.append_protocol_name(EqualityProof::protocol_name());
+
+    // produce a random Scalar
+    let r = random_tape.challenge_scalar(b"r");
+
+    let C1 = v1.commit(&s1, gens_n).compress();
+    C1.append_to_transcript(b"C1", transcript);
+
+    let C2 = v2.commit(&s2, gens_n).compress();
+    C2.append_to_transcript(b"C2", transcript);
+
+    let alpha = (&r * gens_n.h).compress();
+    alpha.append_to_transcript(b"alpha", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z = &c * (s1 - s2) + &r;
+
+    (EqualityProof { alpha, z }, C1, C2)
+  }
+
+  pub fn verify(
+    &self,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    C1: &CompressedGroup,
+    C2: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(EqualityProof::protocol_name());
+    C1.append_to_transcript(b"C1", transcript);
+    C2.append_to_transcript(b"C2", transcript);
+    self.alpha.append_to_transcript(b"alpha", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+    let rhs = {
+      let C = &C1.decompress().unwrap() - &C2.decompress().unwrap();
+      (&c * C + &self.alpha.decompress().unwrap()).compress()
+    };
+
+    let lhs = (&self.z * gens_n.h).compress();
+
+    if lhs == rhs {
+      Ok(())
+    } else {
+      Err(ProofVerifyError)
+    }
+  }
+}
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct ProductProof {
+  alpha: CompressedGroup,
+  beta: CompressedGroup,
+  delta: CompressedGroup,
+  z: [Scalar; 5],
+}
+
+impl ProductProof {
+  fn protocol_name() -> &'static [u8] {
+    b"product proof"
+  }
+
+  pub fn prove(
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+    x: &Scalar,
+    rX: &Scalar,
+    y: &Scalar,
+    rY: &Scalar,
+    z: &Scalar,
+    rZ: &Scalar,
+  ) -> (
+    ProductProof,
+    CompressedGroup,
+    CompressedGroup,
+    CompressedGroup,
+  ) {
+    transcript.append_protocol_name(ProductProof::protocol_name());
+
+    // produce five random Scalar
+    let b1 = random_tape.challenge_scalar(b"b1");
+    let b2 = random_tape.challenge_scalar(b"b2");
+    let b3 = random_tape.challenge_scalar(b"b3");
+    let b4 = random_tape.challenge_scalar(b"b4");
+    let b5 = random_tape.challenge_scalar(b"b5");
+
+    let X = x.commit(&rX, gens_n).compress();
+    X.append_to_transcript(b"X", transcript);
+
+    let Y = y.commit(&rY, gens_n).compress();
+    Y.append_to_transcript(b"Y", transcript);
+
+    let Z = z.commit(&rZ, gens_n).compress();
+    Z.append_to_transcript(b"Z", transcript);
+
+    let alpha = b1.commit(&b2, gens_n).compress();
+    alpha.append_to_transcript(b"alpha", transcript);
+
+    let beta = b3.commit(&b4, gens_n).compress();
+    beta.append_to_transcript(b"beta", transcript);
+
+    let delta = {
+      let gens_X = &MultiCommitGens {
+        n: 1,
+        G: vec![X.decompress().unwrap()],
+        h: gens_n.h,
+      };
+      b3.commit(&b5, gens_X).compress()
+    };
+    delta.append_to_transcript(b"delta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z1 = &b1 + &c * x;
+    let z2 = &b2 + &c * rX;
+    let z3 = &b3 + &c * y;
+    let z4 = &b4 + &c * rY;
+    let z5 = &b5 + &c * (rZ - rX * y);
+    let z = [z1, z2, z3, z4, z5];
+
+    (
+      ProductProof {
+        alpha,
+        beta,
+        delta,
+        z,
+      },
+      X,
+      Y,
+      Z,
+    )
+  }
+
+  fn check_equality(
+    P: &CompressedGroup,
+    X: &CompressedGroup,
+    c: &Scalar,
+    gens_n: &MultiCommitGens,
+    z1: &Scalar,
+    z2: &Scalar,
+  ) -> bool {
+    let lhs = (P.decompress().unwrap() + c * X.decompress().unwrap()).compress();
+    let rhs = z1.commit(&z2, gens_n).compress();
+
+    if lhs == rhs {
+      true
+    } else {
+      false
+    }
+  }
+
+  pub fn verify(
+    &self,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    X: &CompressedGroup,
+    Y: &CompressedGroup,
+    Z: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(ProductProof::protocol_name());
+
+    X.append_to_transcript(b"X", transcript);
+    Y.append_to_transcript(b"Y", transcript);
+    Z.append_to_transcript(b"Z", transcript);
+    self.alpha.append_to_transcript(b"alpha", transcript);
+    self.beta.append_to_transcript(b"beta", transcript);
+    self.delta.append_to_transcript(b"delta", transcript);
+
+    let z1 = self.z[0];
+    let z2 = self.z[1];
+    let z3 = self.z[2];
+    let z4 = self.z[3];
+    let z5 = self.z[4];
+
+    let c = transcript.challenge_scalar(b"c");
+
+    if ProductProof::check_equality(&self.alpha, &X, &c, &gens_n, &z1, &z2)
+      && ProductProof::check_equality(&self.beta, &Y, &c, &gens_n, &z3, &z4)
+      && ProductProof::check_equality(
+        &self.delta,
+        &Z,
+        &c,
+        &MultiCommitGens {
+          n: 1,
+          G: vec![X.decompress().unwrap()],
+          h: gens_n.h,
+        },
+        &z3,
+        &z5,
+      )
+    {
+      Ok(())
+    } else {
+      Err(ProofVerifyError)
+    }
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct DotProductProof {
+  delta: CompressedGroup,
+  beta: CompressedGroup,
+  z: Vec<Scalar>,
+  z_delta: Scalar,
+  z_beta: Scalar,
+}
+
+impl DotProductProof {
+  fn protocol_name() -> &'static [u8] {
+    b"dot product proof"
+  }
+
+  pub fn compute_dotproduct(a: &Vec<Scalar>, b: &Vec<Scalar>) -> Scalar {
+    assert_eq!(a.len(), b.len());
+    (0..a.len()).map(|i| &a[i] * &b[i]).sum()
+  }
+
+  pub fn prove(
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+    x: &Vec<Scalar>,
+    r_x: &Scalar,
+    a: &Vec<Scalar>,
+    y: &Scalar,
+    r_y: &Scalar,
+  ) -> (DotProductProof, CompressedGroup, CompressedGroup) {
+    transcript.append_protocol_name(DotProductProof::protocol_name());
+
+    let n = x.len();
+    assert_eq!(x.len(), a.len());
+    assert_eq!(gens_n.n, a.len());
+    assert_eq!(gens_1.n, 1);
+
+    // produce randomness for the proofs
+    let d = random_tape.challenge_vector(b"d", n);
+    let r_delta = random_tape.challenge_scalar(b"r_delta");
+    let r_beta = random_tape.challenge_scalar(b"r_beta");
+
+    let Cx = x.commit(&r_x, gens_n).compress();
+    Cx.append_to_transcript(b"Cx", transcript);
+
+    let Cy = y.commit(&r_y, gens_1).compress();
+    Cy.append_to_transcript(b"Cy", transcript);
+
+    let delta = d.commit(&r_delta, gens_n).compress();
+    delta.append_to_transcript(b"delta", transcript);
+
+    let dotproduct_a_d = DotProductProof::compute_dotproduct(&a, &d);
+
+    let beta = dotproduct_a_d.commit(&r_beta, gens_1).compress();
+    beta.append_to_transcript(b"beta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z = (0..d.len())
+      .map(|i| c * x[i] + d[i])
+      .collect::<Vec<Scalar>>();
+
+    let z_delta = c * r_x + r_delta;
+    let z_beta = c * r_y + r_beta;
+
+    (
+      DotProductProof {
+        delta,
+        beta,
+        z,
+        z_delta,
+        z_beta,
+      },
+      Cx,
+      Cy,
+    )
+  }
+
+  pub fn verify(
+    &self,
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    a: &Vec<Scalar>,
+    Cx: &CompressedGroup,
+    Cy: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    assert_eq!(gens_n.n, a.len());
+    assert_eq!(gens_1.n, 1);
+
+    transcript.append_protocol_name(DotProductProof::protocol_name());
+    Cx.append_to_transcript(b"Cx", transcript);
+    Cy.append_to_transcript(b"Cy", transcript);
+    self.delta.append_to_transcript(b"delta", transcript);
+    self.beta.append_to_transcript(b"beta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let mut result = &c * Cx.decompress().unwrap() + self.delta.decompress().unwrap()
+      == self.z.commit(&self.z_delta, gens_n);
+
+    let dotproduct_z_a = DotProductProof::compute_dotproduct(&self.z, &a);
+    result &= &c * Cy.decompress().unwrap() + self.beta.decompress().unwrap()
+      == dotproduct_z_a.commit(&self.z_beta, gens_1);
+
+    if result {
+      Ok(())
+    } else {
+      Err(ProofVerifyError)
+    }
+  }
+}
+
+pub struct DotProductProofGens {
+  n: usize,
+  pub gens_n: MultiCommitGens,
+  pub gens_1: MultiCommitGens,
+}
+
+impl DotProductProofGens {
+  pub fn new(n: usize, label: &[u8]) -> Self {
+    let (gens_n, gens_1) = MultiCommitGens::new(n + 1, label).split_at_mut(n);
+    DotProductProofGens { n, gens_n, gens_1 }
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct DotProductProofLog {
+  bullet_reduction_proof: BulletReductionProof,
+  delta: CompressedGroup,
+  beta: CompressedGroup,
+  z1: Scalar,
+  z2: Scalar,
+}
+
+impl DotProductProofLog {
+  fn protocol_name() -> &'static [u8] {
+    b"dot product proof (log)"
+  }
+
+  pub fn compute_dotproduct(a: &Vec<Scalar>, b: &Vec<Scalar>) -> Scalar {
+    assert_eq!(a.len(), b.len());
+    (0..a.len()).map(|i| &a[i] * &b[i]).sum()
+  }
+
+  pub fn prove(
+    gens: &DotProductProofGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+    x: &Vec<Scalar>,
+    r_x: &Scalar,
+    a: &Vec<Scalar>,
+    y: &Scalar,
+    r_y: &Scalar,
+  ) -> (DotProductProofLog, CompressedGroup, CompressedGroup) {
+    transcript.append_protocol_name(DotProductProofLog::protocol_name());
+
+    let n = x.len();
+    assert_eq!(x.len(), a.len());
+    assert_eq!(gens.n, n);
+
+    // produce randomness for generating a proof
+    let d = random_tape.challenge_scalar(b"d");
+    let r_delta = random_tape.challenge_scalar(b"r_delta");
+    let r_beta = random_tape.challenge_scalar(b"r_delta");
+    let blinds_vec = {
+      let v1 = random_tape.challenge_vector(b"blinds_vec_1", 2 * n.log2());
+      let v2 = random_tape.challenge_vector(b"blinds_vec_2", 2 * n.log2());
+      (0..v1.len())
+        .map(|i| (v1[i], v2[i]))
+        .collect::<Vec<(Scalar, Scalar)>>()
+    };
+
+    let Cx = x.commit(&r_x, &gens.gens_n).compress();
+    Cx.append_to_transcript(b"Cx", transcript);
+
+    let Cy = y.commit(&r_y, &gens.gens_1).compress();
+    Cy.append_to_transcript(b"Cy", transcript);
+
+    let r_Gamma = r_x + r_y;
+    let (bullet_reduction_proof, _Gamma_hat, x_hat, a_hat, g_hat, rhat_Gamma) =
+      BulletReductionProof::prove(
+        transcript,
+        &gens.gens_1.G[0],
+        &gens.gens_n.G,
+        &gens.gens_n.h,
+        x,
+        a,
+        &r_Gamma,
+        &blinds_vec,
+      );
+    let y_hat = x_hat * a_hat;
+
+    let delta = {
+      let gens_hat = MultiCommitGens {
+        n: 1,
+        G: vec![g_hat],
+        h: gens.gens_1.h,
+      };
+      d.commit(&r_delta, &gens_hat).compress()
+    };
+    delta.append_to_transcript(b"delta", transcript);
+
+    let beta = d.commit(&r_beta, &gens.gens_1).compress();
+    beta.append_to_transcript(b"beta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z1 = d + c * y_hat;
+    let z2 = a_hat * (c * rhat_Gamma + r_beta) + r_delta;
+
+    (
+      DotProductProofLog {
+        bullet_reduction_proof,
+        delta,
+        beta,
+        z1,
+        z2,
+      },
+      Cx,
+      Cy,
+    )
+  }
+
+  pub fn verify(
+    &self,
+    n: usize,
+    gens: &DotProductProofGens,
+    transcript: &mut Transcript,
+    a: &Vec<Scalar>,
+    Cx: &CompressedGroup,
+    Cy: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    assert_eq!(gens.n, n);
+    assert_eq!(a.len(), n);
+
+    transcript.append_protocol_name(DotProductProofLog::protocol_name());
+    Cx.append_to_transcript(b"Cx", transcript);
+    Cy.append_to_transcript(b"Cy", transcript);
+
+    let Gamma = Cx.decompress().unwrap() + Cy.decompress().unwrap();
+
+    let (g_hat, Gamma_hat, a_hat) = self
+      .bullet_reduction_proof
+      .verify(n, a, transcript, &Gamma, &gens.gens_n.G)
+      .unwrap();
+
+    self.delta.append_to_transcript(b"delta", transcript);
+    self.beta.append_to_transcript(b"beta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let c_s = &c;
+    let beta_s = self.beta.decompress().unwrap();
+    let a_hat_s = &a_hat;
+    let delta_s = self.delta.decompress().unwrap();
+    let z1_s = &self.z1;
+    let z2_s = &self.z2;
+
+    let lhs = ((Gamma_hat * c_s + beta_s) * a_hat_s + delta_s).compress();
+    let rhs = ((g_hat + &gens.gens_1.G[0] * a_hat_s) * z1_s + gens.gens_1.h * z2_s).compress();
+
+    assert_eq!(lhs, rhs);
+
+    if lhs == rhs {
+      Ok(())
+    } else {
+      Err(ProofVerifyError)
+    }
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+  use rand::rngs::OsRng;
+  #[test]
+  fn check_knowledgeproof() {
+    let mut csprng: OsRng = OsRng;
+
+    let gens_1 = MultiCommitGens::new(1, b"test-knowledgeproof");
+
+    let x = Scalar::random(&mut csprng);
+    let r = Scalar::random(&mut csprng);
+
+    let mut random_tape = {
+      let mut csprng: OsRng = OsRng;
+      let mut tape = Transcript::new(b"proof");
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut csprng));
+      tape
+    };
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, committed_value) =
+      KnowledgeProof::prove(&gens_1, &mut prover_transcript, &mut random_tape, &x, &r);
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens_1, &mut verifier_transcript, &committed_value)
+      .is_ok());
+  }
+
+  #[test]
+  fn check_equalityproof() {
+    let mut csprng: OsRng = OsRng;
+
+    let gens_1 = MultiCommitGens::new(1, b"test-equalityproof");
+    let v1 = Scalar::random(&mut csprng);
+    let v2 = v1;
+    let s1 = Scalar::random(&mut csprng);
+    let s2 = Scalar::random(&mut csprng);
+
+    let mut random_tape = {
+      let mut csprng: OsRng = OsRng;
+      let mut tape = Transcript::new(b"proof");
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut csprng));
+      tape
+    };
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, C1, C2) = EqualityProof::prove(
+      &gens_1,
+      &mut prover_transcript,
+      &mut random_tape,
+      &v1,
+      &s1,
+      &v2,
+      &s2,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens_1, &mut verifier_transcript, &C1, &C2)
+      .is_ok());
+  }
+
+  #[test]
+  fn check_productproof() {
+    let mut csprng: OsRng = OsRng;
+
+    let gens_1 = MultiCommitGens::new(1, b"test-productproof");
+    let x = Scalar::random(&mut csprng);
+    let rX = Scalar::random(&mut csprng);
+    let y = Scalar::random(&mut csprng);
+    let rY = Scalar::random(&mut csprng);
+    let z = x * y;
+    let rZ = Scalar::random(&mut csprng);
+
+    let mut random_tape = {
+      let mut csprng: OsRng = OsRng;
+      let mut tape = Transcript::new(b"proof");
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut csprng));
+      tape
+    };
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, X, Y, Z) = ProductProof::prove(
+      &gens_1,
+      &mut prover_transcript,
+      &mut random_tape,
+      &x,
+      &rX,
+      &y,
+      &rY,
+      &z,
+      &rZ,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens_1, &mut verifier_transcript, &X, &Y, &Z)
+      .is_ok());
+  }
+
+  #[test]
+  fn check_dotproductproof() {
+    let mut csprng: OsRng = OsRng;
+
+    let n = 1024;
+
+    let gens_1 = MultiCommitGens::new(1, b"test-two");
+    let gens_1024 = MultiCommitGens::new(n, b"test-1024");
+
+    let mut x: Vec<Scalar> = Vec::new();
+    let mut a: Vec<Scalar> = Vec::new();
+    for _ in 0..n {
+      x.push(Scalar::random(&mut csprng));
+      a.push(Scalar::random(&mut csprng));
+    }
+    let y = DotProductProofLog::compute_dotproduct(&x, &a);
+    let r_x = Scalar::random(&mut csprng);
+    let r_y = Scalar::random(&mut csprng);
+
+    let mut random_tape = {
+      let mut csprng: OsRng = OsRng;
+      let mut tape = Transcript::new(b"proof");
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut csprng));
+      tape
+    };
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, Cx, Cy) = DotProductProof::prove(
+      &gens_1,
+      &gens_1024,
+      &mut prover_transcript,
+      &mut random_tape,
+      &x,
+      &r_x,
+      &a,
+      &y,
+      &r_y,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens_1, &gens_1024, &mut verifier_transcript, &a, &Cx, &Cy)
+      .is_ok());
+  }
+
+  #[test]
+  fn check_dotproductproof_log() {
+    let mut csprng: OsRng = OsRng;
+
+    let n = 1024;
+
+    let gens = DotProductProofGens::new(n, b"test-1024");
+
+    let x: Vec<Scalar> = (0..n).map(|_i| Scalar::random(&mut csprng)).collect();
+    let a: Vec<Scalar> = (0..n).map(|_i| Scalar::random(&mut csprng)).collect();
+    let y = DotProductProof::compute_dotproduct(&x, &a);
+
+    let r_x = Scalar::random(&mut csprng);
+    let r_y = Scalar::random(&mut csprng);
+
+    let mut random_tape = {
+      let mut csprng: OsRng = OsRng;
+      let mut tape = Transcript::new(b"proof");
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut csprng));
+      tape
+    };
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, Cx, Cy) = DotProductProofLog::prove(
+      &gens,
+      &mut prover_transcript,
+      &mut random_tape,
+      &x,
+      &r_x,
+      &a,
+      &y,
+      &r_y,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(n, &gens, &mut verifier_transcript, &a, &Cx, &Cy)
+      .is_ok());
+  }
+}

src/product_tree.rs

@@ -0,0 +1,489 @@
+#[allow(dead_code)]
+use super::dense_mlpoly::DensePolynomial;
+use super::dense_mlpoly::EqPolynomial;
+use super::math::Math;
+use super::scalar::Scalar;
+use super::sumcheck::SumcheckInstanceProof;
+use super::transcript::ProofTranscript;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+#[derive(Debug)]
+pub struct ProductCircuit {
+  left_vec: Vec<DensePolynomial>,
+  right_vec: Vec<DensePolynomial>,
+}
+
+impl ProductCircuit {
+  fn compute_layer(
+    inp_left: &DensePolynomial,
+    inp_right: &DensePolynomial,
+  ) -> (DensePolynomial, DensePolynomial) {
+    let len = inp_left.len() + inp_right.len();
+    let outp_left = (0..len / 4)
+      .map(|i| &inp_left[i] * &inp_right[i])
+      .collect::<Vec<Scalar>>();
+    let outp_right = (len / 4..len / 2)
+      .map(|i| &inp_left[i] * &inp_right[i])
+      .collect::<Vec<Scalar>>();
+
+    (
+      DensePolynomial::new(outp_left),
+      DensePolynomial::new(outp_right),
+    )
+  }
+
+  pub fn new(poly: &DensePolynomial) -> Self {
+    let mut left_vec: Vec<DensePolynomial> = Vec::new();
+    let mut right_vec: Vec<DensePolynomial> = Vec::new();
+
+    let num_layers = poly.len().log2();
+    let (outp_left, outp_right) = poly.split(poly.len() / 2);
+
+    left_vec.push(outp_left);
+    right_vec.push(outp_right);
+
+    for i in 0..num_layers - 1 {
+      let (outp_left, outp_right) = ProductCircuit::compute_layer(&left_vec[i], &right_vec[i]);
+      left_vec.push(outp_left);
+      right_vec.push(outp_right);
+    }
+
+    ProductCircuit {
+      left_vec,
+      right_vec,
+    }
+  }
+
+  pub fn evaluate(&self) -> Scalar {
+    let len = self.left_vec.len();
+    assert_eq!(self.left_vec[len - 1].get_num_vars(), 0);
+    assert_eq!(self.right_vec[len - 1].get_num_vars(), 0);
+    self.left_vec[len - 1][0] * self.right_vec[len - 1][0]
+  }
+}
+
+pub struct DotProductCircuit {
+  left: DensePolynomial,
+  right: DensePolynomial,
+  weight: DensePolynomial,
+}
+
+impl DotProductCircuit {
+  pub fn new(left: DensePolynomial, right: DensePolynomial, weight: DensePolynomial) -> Self {
+    assert_eq!(left.len(), right.len());
+    assert_eq!(left.len(), weight.len());
+    DotProductCircuit {
+      left,
+      right,
+      weight,
+    }
+  }
+
+  pub fn evaluate(&self) -> Scalar {
+    (0..self.left.len())
+      .map(|i| &self.left[i] * &self.right[i] * &self.weight[i])
+      .sum()
+  }
+
+  pub fn split(&mut self) -> (DotProductCircuit, DotProductCircuit) {
+    let idx = self.left.len() / 2;
+    assert_eq!(idx * 2, self.left.len());
+    let (l1, l2) = self.left.split(idx);
+    let (r1, r2) = self.right.split(idx);
+    let (w1, w2) = self.weight.split(idx);
+    (
+      DotProductCircuit {
+        left: l1,
+        right: r1,
+        weight: w1,
+      },
+      DotProductCircuit {
+        left: l2,
+        right: r2,
+        weight: w2,
+      },
+    )
+  }
+}
+
+#[allow(dead_code)]
+#[derive(Debug, Serialize, Deserialize)]
+pub struct LayerProof {
+  pub proof: SumcheckInstanceProof,
+  pub claims: Vec<Scalar>,
+}
+
+#[allow(dead_code)]
+impl LayerProof {
+  pub fn verify(
+    &self,
+    claim: Scalar,
+    num_rounds: usize,
+    degree_bound: usize,
+    transcript: &mut Transcript,
+  ) -> (Scalar, Vec<Scalar>) {
+    self
+      .proof
+      .verify(claim, num_rounds, degree_bound, transcript)
+      .unwrap()
+  }
+}
+
+#[allow(dead_code)]
+#[derive(Debug, Serialize, Deserialize)]
+pub struct LayerProofBatched {
+  pub proof: SumcheckInstanceProof,
+  pub claims_prod_left: Vec<Scalar>,
+  pub claims_prod_right: Vec<Scalar>,
+}
+
+#[allow(dead_code)]
+impl LayerProofBatched {
+  pub fn verify(
+    &self,
+    claim: Scalar,
+    num_rounds: usize,
+    degree_bound: usize,
+    transcript: &mut Transcript,
+  ) -> (Scalar, Vec<Scalar>) {
+    self
+      .proof
+      .verify(claim, num_rounds, degree_bound, transcript)
+      .unwrap()
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct ProductCircuitEvalProof {
+  proof: Vec<LayerProof>,
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct ProductCircuitEvalProofBatched {
+  proof: Vec<LayerProofBatched>,
+  claims_dotp: (Vec<Scalar>, Vec<Scalar>, Vec<Scalar>),
+}
+
+impl ProductCircuitEvalProof {
+  #![allow(dead_code)]
+  pub fn prove(
+    circuit: &mut ProductCircuit,
+    transcript: &mut Transcript,
+  ) -> (Self, Scalar, Vec<Scalar>) {
+    let mut proof: Vec<LayerProof> = Vec::new();
+    let num_layers = circuit.left_vec.len();
+
+    let mut claim = circuit.evaluate();
+    let mut rand = Vec::new();
+    for layer_id in (0..num_layers).rev() {
+      let len = circuit.left_vec[layer_id].len() + circuit.right_vec[layer_id].len();
+
+      let mut poly_C = DensePolynomial::new(EqPolynomial::new(rand.clone()).evals());
+      assert_eq!(poly_C.len(), len / 2);
+
+      let num_rounds_prod = poly_C.len().log2();
+      let comb_func_prod = |poly_A_comp: &Scalar,
+                            poly_B_comp: &Scalar,
+                            poly_C_comp: &Scalar|
+       -> Scalar { poly_A_comp * poly_B_comp * poly_C_comp };
+      let (proof_prod, rand_prod, claims_prod) = SumcheckInstanceProof::prove_cubic(
+        &claim,
+        num_rounds_prod,
+        &mut circuit.left_vec[layer_id],
+        &mut circuit.right_vec[layer_id],
+        &mut poly_C,
+        comb_func_prod,
+        transcript,
+      );
+
+      transcript.append_scalar(b"claim_prod_left", &claims_prod[0]);
+      transcript.append_scalar(b"claim_prod_right", &claims_prod[1]);
+
+      // produce a random challenge
+      let r_layer = transcript.challenge_scalar(b"challenge_r_layer");
+      claim = &claims_prod[0] + &r_layer * (&claims_prod[1] - &claims_prod[0]);
+
+      let mut ext = vec![r_layer];
+      ext.extend(rand_prod);
+      rand = ext;
+
+      proof.push(LayerProof {
+        proof: proof_prod,
+        claims: claims_prod[0..claims_prod.len() - 1].to_vec(),
+      });
+    }
+
+    (ProductCircuitEvalProof { proof }, claim, rand)
+  }
+
+  pub fn verify(
+    &self,
+    eval: Scalar,
+    len: usize,
+    transcript: &mut Transcript,
+  ) -> (Scalar, Vec<Scalar>) {
+    let num_layers = len.log2();
+    let mut claim = eval;
+    let mut rand: Vec<Scalar> = Vec::new();
+    let mut num_rounds = 0;
+    assert_eq!(self.proof.len(), num_layers);
+    for i in 0..num_layers {
+      let (claim_last, rand_prod) = self.proof[i].verify(claim, num_rounds, 3, transcript);
+
+      let claims_prod = &self.proof[i].claims;
+      transcript.append_scalar(b"claim_prod_left", &claims_prod[0]);
+      transcript.append_scalar(b"claim_prod_right", &claims_prod[1]);
+
+      assert_eq!(rand.len(), rand_prod.len());
+      let eq: Scalar = (0..rand.len())
+        .map(|i| {
+          rand[i] * rand_prod[i] + (Scalar::one() - rand[i]) * (Scalar::one() - rand_prod[i])
+        })
+        .product();
+      assert_eq!(claims_prod[0] * claims_prod[1] * eq, claim_last);
+
+      // produce a random challenge
+      let r_layer = transcript.challenge_scalar(b"challenge_r_layer");
+      claim = (Scalar::one() - r_layer) * claims_prod[0] + r_layer * claims_prod[1];
+      num_rounds = num_rounds + 1;
+      let mut ext = vec![r_layer];
+      ext.extend(rand_prod);
+      rand = ext;
+    }
+
+    (claim, rand)
+  }
+}
+
+impl ProductCircuitEvalProofBatched {
+  pub fn prove(
+    prod_circuit_vec: &mut Vec<&mut ProductCircuit>,
+    dotp_circuit_vec: &mut Vec<&mut DotProductCircuit>,
+    transcript: &mut Transcript,
+  ) -> (Self, Vec<Scalar>) {
+    assert!(prod_circuit_vec.len() > 0);
+
+    let mut claims_dotp_final = (Vec::new(), Vec::new(), Vec::new());
+
+    let mut proof_layers: Vec<LayerProofBatched> = Vec::new();
+    let num_layers = prod_circuit_vec[0].left_vec.len();
+    let mut claims_to_verify = (0..prod_circuit_vec.len())
+      .map(|i| prod_circuit_vec[i].evaluate())
+      .collect::<Vec<Scalar>>();
+    let mut rand = Vec::new();
+    for layer_id in (0..num_layers).rev() {
+      // prepare paralell instance that share poly_C first
+      let len = prod_circuit_vec[0].left_vec[layer_id].len()
+        + prod_circuit_vec[0].right_vec[layer_id].len();
+
+      let mut poly_C_par = DensePolynomial::new(EqPolynomial::new(rand.clone()).evals());
+      assert_eq!(poly_C_par.len(), len / 2);
+
+      let num_rounds_prod = poly_C_par.len().log2();
+      let comb_func_prod = |poly_A_comp: &Scalar,
+                            poly_B_comp: &Scalar,
+                            poly_C_comp: &Scalar|
+       -> Scalar { poly_A_comp * poly_B_comp * poly_C_comp };
+
+      let mut poly_A_batched_par: Vec<&mut DensePolynomial> = Vec::new();
+      let mut poly_B_batched_par: Vec<&mut DensePolynomial> = Vec::new();
+      for prod_circuit in prod_circuit_vec.iter_mut() {
+        poly_A_batched_par.push(&mut prod_circuit.left_vec[layer_id]);
+        poly_B_batched_par.push(&mut prod_circuit.right_vec[layer_id])
+      }
+      let poly_vec_par = (
+        &mut poly_A_batched_par,
+        &mut poly_B_batched_par,
+        &mut poly_C_par,
+      );
+
+      // prepare sequential instances that don't share poly_C
+      let mut poly_A_batched_seq: Vec<&mut DensePolynomial> = Vec::new();
+      let mut poly_B_batched_seq: Vec<&mut DensePolynomial> = Vec::new();
+      let mut poly_C_batched_seq: Vec<&mut DensePolynomial> = Vec::new();
+      if layer_id == 0 && dotp_circuit_vec.len() > 0 {
+        // add additional claims
+        for i in 0..dotp_circuit_vec.len() {
+          claims_to_verify.push(dotp_circuit_vec[i].evaluate());
+          assert_eq!(len / 2, dotp_circuit_vec[i].left.len());
+          assert_eq!(len / 2, dotp_circuit_vec[i].right.len());
+          assert_eq!(len / 2, dotp_circuit_vec[i].weight.len());
+        }
+
+        for dotp_circuit in dotp_circuit_vec.iter_mut() {
+          poly_A_batched_seq.push(&mut dotp_circuit.left);
+          poly_B_batched_seq.push(&mut dotp_circuit.right);
+          poly_C_batched_seq.push(&mut dotp_circuit.weight);
+        }
+      }
+      let poly_vec_seq = (
+        &mut poly_A_batched_seq,
+        &mut poly_B_batched_seq,
+        &mut poly_C_batched_seq,
+      );
+
+      // produce a fresh set of coeffs and a joint claim
+      let coeff_vec =
+        transcript.challenge_vector(b"rand_coeffs_next_layer", claims_to_verify.len());
+      let claim = (0..claims_to_verify.len())
+        .map(|i| claims_to_verify[i] * coeff_vec[i])
+        .sum();
+
+      let (proof, rand_prod, claims_prod, claims_dotp) = SumcheckInstanceProof::prove_cubic_batched(
+        &claim,
+        num_rounds_prod,
+        poly_vec_par,
+        poly_vec_seq,
+        &coeff_vec,
+        comb_func_prod,
+        transcript,
+      );
+
+      let (claims_prod_left, claims_prod_right, _claims_eq) = claims_prod;
+      for i in 0..prod_circuit_vec.len() {
+        transcript.append_scalar(b"claim_prod_left", &claims_prod_left[i]);
+        transcript.append_scalar(b"claim_prod_right", &claims_prod_right[i]);
+      }
+
+      if layer_id == 0 && dotp_circuit_vec.len() > 0 {
+        let (claims_dotp_left, claims_dotp_right, claims_dotp_weight) = claims_dotp;
+        for i in 0..dotp_circuit_vec.len() {
+          transcript.append_scalar(b"claim_dotp_left", &claims_dotp_left[i]);
+          transcript.append_scalar(b"claim_dotp_right", &claims_dotp_right[i]);
+          transcript.append_scalar(b"claim_dotp_weight", &claims_dotp_weight[i]);
+        }
+        claims_dotp_final = (claims_dotp_left, claims_dotp_right, claims_dotp_weight);
+      }
+
+      // produce a random challenge to condense two claims into a single claim
+      let r_layer = transcript.challenge_scalar(b"challenge_r_layer");
+
+      claims_to_verify = (0..prod_circuit_vec.len())
+        .map(|i| &claims_prod_left[i] + &r_layer * (&claims_prod_right[i] - &claims_prod_left[i]))
+        .collect::<Vec<Scalar>>();
+
+      let mut ext = vec![r_layer];
+      ext.extend(rand_prod);
+      rand = ext;
+
+      proof_layers.push(LayerProofBatched {
+        proof,
+        claims_prod_left,
+        claims_prod_right,
+      });
+    }
+
+    (
+      ProductCircuitEvalProofBatched {
+        proof: proof_layers,
+        claims_dotp: claims_dotp_final,
+      },
+      rand,
+    )
+  }
+
+  pub fn verify(
+    &self,
+    claims_prod_vec: &Vec<Scalar>,
+    claims_dotp_vec: &Vec<Scalar>,
+    len: usize,
+    transcript: &mut Transcript,
+  ) -> (Vec<Scalar>, Vec<Scalar>, Vec<Scalar>) {
+    let num_layers = len.log2();
+    let mut rand: Vec<Scalar> = Vec::new();
+    let mut num_rounds = 0;
+    assert_eq!(self.proof.len(), num_layers);
+
+    let mut claims_to_verify = claims_prod_vec.clone();
+    let mut claims_to_verify_dotp: Vec<Scalar> = Vec::new();
+    for i in 0..num_layers {
+      if i == num_layers - 1 {
+        claims_to_verify.extend(claims_dotp_vec);
+      }
+
+      // produce random coefficients, one for each instance
+      let coeff_vec =
+        transcript.challenge_vector(b"rand_coeffs_next_layer", claims_to_verify.len());
+
+      // produce a joint claim
+      let claim = (0..claims_to_verify.len())
+        .map(|i| claims_to_verify[i] * coeff_vec[i])
+        .sum();
+
+      let (claim_last, rand_prod) = self.proof[i].verify(claim, num_rounds, 3, transcript);
+
+      let claims_prod_left = &self.proof[i].claims_prod_left;
+      let claims_prod_right = &self.proof[i].claims_prod_right;
+      assert_eq!(claims_prod_left.len(), claims_prod_vec.len());
+      assert_eq!(claims_prod_right.len(), claims_prod_vec.len());
+
+      for i in 0..claims_prod_vec.len() {
+        transcript.append_scalar(b"claim_prod_left", &claims_prod_left[i]);
+        transcript.append_scalar(b"claim_prod_right", &claims_prod_right[i]);
+      }
+
+      assert_eq!(rand.len(), rand_prod.len());
+      let eq: Scalar = (0..rand.len())
+        .map(|i| {
+          rand[i] * rand_prod[i] + (Scalar::one() - rand[i]) * (Scalar::one() - rand_prod[i])
+        })
+        .product();
+      let mut claim_expected: Scalar = (0..claims_prod_vec.len())
+        .map(|i| coeff_vec[i] * (claims_prod_left[i] * claims_prod_right[i] * eq))
+        .sum();
+
+      // add claims from the dotp instances
+      if i == num_layers - 1 {
+        let num_prod_instances = claims_prod_vec.len();
+        let (claims_dotp_left, claims_dotp_right, claims_dotp_weight) = &self.claims_dotp;
+        for i in 0..claims_dotp_left.len() {
+          transcript.append_scalar(b"claim_dotp_left", &claims_dotp_left[i]);
+          transcript.append_scalar(b"claim_dotp_right", &claims_dotp_right[i]);
+          transcript.append_scalar(b"claim_dotp_weight", &claims_dotp_weight[i]);
+
+          claim_expected = &claim_expected
+            + &coeff_vec[i + num_prod_instances]
+              * &claims_dotp_left[i]
+              * &claims_dotp_right[i]
+              * &claims_dotp_weight[i];
+        }
+      }
+
+      assert_eq!(claim_expected, claim_last);
+
+      // produce a random challenge
+      let r_layer = transcript.challenge_scalar(b"challenge_r_layer");
+
+      claims_to_verify = (0..claims_prod_left.len())
+        .map(|i| &claims_prod_left[i] + &r_layer * (&claims_prod_right[i] - &claims_prod_left[i]))
+        .collect::<Vec<Scalar>>();
+
+      // add claims to verify for dotp circuit
+      if i == num_layers - 1 {
+        let (claims_dotp_left, claims_dotp_right, claims_dotp_weight) = &self.claims_dotp;
+
+        for i in 0..claims_dotp_vec.len() / 2 {
+          // combine left claims
+          let claim_left = &claims_dotp_left[2 * i]
+            + &r_layer * (&claims_dotp_left[2 * i + 1] - &claims_dotp_left[2 * i]);
+
+          let claim_right = &claims_dotp_right[2 * i]
+            + &r_layer * (&claims_dotp_right[2 * i + 1] - &claims_dotp_right[2 * i]);
+
+          let claim_weight = &claims_dotp_weight[2 * i]
+            + &r_layer * (&claims_dotp_weight[2 * i + 1] - &claims_dotp_weight[2 * i]);
+          claims_to_verify_dotp.push(claim_left);
+          claims_to_verify_dotp.push(claim_right);
+          claims_to_verify_dotp.push(claim_weight);
+        }
+      }
+
+      num_rounds = num_rounds + 1;
+      let mut ext = vec![r_layer];
+      ext.extend(rand_prod);
+      rand = ext;
+    }
+    (claims_to_verify, claims_to_verify_dotp, rand)
+  }
+}

src/profiler.rs

@@ -0,0 +1,55 @@
+#![allow(non_snake_case)]
+extern crate flate2;
+extern crate libspartan;
+extern crate merlin;
+extern crate rand;
+
+use flate2::{write::ZlibEncoder, Compression};
+use libspartan::math::Math;
+use libspartan::r1csinstance::{R1CSCommitmentGens, R1CSInstance};
+use libspartan::r1csproof::R1CSGens;
+use libspartan::spartan::{SpartanGens, SpartanProof};
+use libspartan::timer::Timer;
+use merlin::Transcript;
+
+pub fn main() {
+  for &s in [12, 16, 20].iter() {
+    let num_vars = (s as usize).pow2();
+    let num_cons = num_vars;
+    let num_inputs = 10;
+    let (inst, vars, input) = R1CSInstance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    let r1cs_size = inst.size();
+    let gens_r1cs_eval = R1CSCommitmentGens::new(&r1cs_size, b"gens_r1cs_eval");
+
+    Timer::print(&format!("number_of_constraints {}", num_cons));
+    // create a commitment to R1CSInstance
+    let timer_encode = Timer::new("SpartanProof::encode");
+    let (comm, decomm) = SpartanProof::encode(&inst, &gens_r1cs_eval);
+    timer_encode.stop();
+
+    let gens_r1cs_sat = R1CSGens::new(num_cons, num_vars, b"gens_r1cs_sat");
+    let gens = SpartanGens::new(gens_r1cs_sat, gens_r1cs_eval);
+
+    // produce a proof of satisfiability
+    let timer_prove = Timer::new("SpartanProof::prove");
+    let mut prover_transcript = Transcript::new(b"example");
+    let proof = SpartanProof::prove(&inst, &decomm, vars, &input, &gens, &mut prover_transcript);
+    timer_prove.stop();
+
+    let mut encoder = ZlibEncoder::new(Vec::new(), Compression::default());
+    bincode::serialize_into(&mut encoder, &proof).unwrap();
+    let proof_encoded = encoder.finish().unwrap();
+    let msg_proof_len = format!("proof_compressed_len {:?}", proof_encoded.len());
+    Timer::print(&msg_proof_len);
+
+    let timer_verify = Timer::new("SpartanProof::verify");
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&comm, &input, &mut verifier_transcript, &gens)
+      .is_ok());
+    timer_verify.stop();
+
+    println!();
+  }
+}

src/r1csinstance.rs

@@ -0,0 +1,346 @@
+use super::dense_mlpoly::DensePolynomial;
+use super::errors::ProofVerifyError;
+use super::math::Math;
+use super::scalar::Scalar;
+use super::sparse_mlpoly::{
+  MultiSparseMatPolynomialAsDense, SparseMatEntry, SparseMatPolyCommitment,
+  SparseMatPolyCommitmentGens, SparseMatPolyEvalProof, SparseMatPolynomial,
+  SparseMatPolynomialSize,
+};
+use super::timer::Timer;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use merlin::Transcript;
+use rand::rngs::OsRng;
+use serde::{Deserialize, Serialize};
+
+#[derive(Debug)]
+pub struct R1CSInstance {
+  num_cons: usize,
+  num_vars: usize,
+  num_inputs: usize,
+  A: SparseMatPolynomial,
+  B: SparseMatPolynomial,
+  C: SparseMatPolynomial,
+}
+
+pub struct R1CSInstanceSize {
+  size_A: SparseMatPolynomialSize,
+  size_B: SparseMatPolynomialSize,
+  size_C: SparseMatPolynomialSize,
+}
+
+pub struct R1CSCommitmentGens {
+  gens: SparseMatPolyCommitmentGens,
+}
+
+impl R1CSCommitmentGens {
+  pub fn new(size: &R1CSInstanceSize, label: &'static [u8]) -> R1CSCommitmentGens {
+    assert_eq!(size.size_A, size.size_B);
+    assert_eq!(size.size_A, size.size_C);
+    let gens = SparseMatPolyCommitmentGens::new(&size.size_A, 3, label);
+    R1CSCommitmentGens { gens }
+  }
+}
+
+pub struct R1CSCommitment {
+  num_cons: usize,
+  num_vars: usize,
+  num_inputs: usize,
+  comm: SparseMatPolyCommitment,
+}
+
+pub struct R1CSDecommitment {
+  dense: MultiSparseMatPolynomialAsDense,
+}
+
+impl R1CSCommitment {
+  pub fn get_num_cons(&self) -> usize {
+    self.num_cons
+  }
+
+  pub fn get_num_vars(&self) -> usize {
+    self.num_vars
+  }
+
+  pub fn get_num_inputs(&self) -> usize {
+    self.num_inputs
+  }
+}
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct R1CSInstanceEvals {
+  eval_A_r: Scalar,
+  eval_B_r: Scalar,
+  eval_C_r: Scalar,
+}
+
+impl R1CSInstanceEvals {
+  pub fn get_evaluations(&self) -> (Scalar, Scalar, Scalar) {
+    (self.eval_A_r, self.eval_B_r, self.eval_C_r)
+  }
+}
+
+impl AppendToTranscript for R1CSInstanceEvals {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_message(label, b"R1CSInstanceEvals_begin");
+    transcript.append_scalar(b"Ar_eval", &self.eval_A_r);
+    transcript.append_scalar(b"Br_eval", &self.eval_B_r);
+    transcript.append_scalar(b"Cr_eval", &self.eval_C_r);
+    transcript.append_message(label, b"R1CSInstanceEvals_end");
+  }
+}
+
+impl R1CSInstance {
+  pub fn new(
+    num_cons: usize,
+    num_vars: usize,
+    num_inputs: usize,
+    A: SparseMatPolynomial,
+    B: SparseMatPolynomial,
+    C: SparseMatPolynomial,
+  ) -> Self {
+    R1CSInstance {
+      num_cons,
+      num_vars,
+      num_inputs,
+      A,
+      B,
+      C,
+    }
+  }
+
+  pub fn get_num_vars(&self) -> usize {
+    self.num_vars
+  }
+
+  pub fn get_num_cons(&self) -> usize {
+    self.num_cons
+  }
+
+  pub fn size(&self) -> R1CSInstanceSize {
+    R1CSInstanceSize {
+      size_A: self.A.size(),
+      size_B: self.B.size(),
+      size_C: self.C.size(),
+    }
+  }
+
+  pub fn produce_synthetic_r1cs(
+    num_cons: usize,
+    num_vars: usize,
+    num_inputs: usize,
+  ) -> (R1CSInstance, Vec<Scalar>, Vec<Scalar>) {
+    let mut csprng: OsRng = OsRng;
+
+    // assert num_cons and num_vars are power of 2
+    assert_eq!(num_cons.log2().pow2(), num_cons);
+    assert_eq!(num_vars.log2().pow2(), num_vars);
+
+    // num_inputs + 1 <= num_vars
+    assert!(num_inputs + 1 <= num_vars);
+
+    // z is organized as [vars,1,io]
+    let size_z = num_vars + num_inputs + 1;
+
+    // produce a random satisfying assignment
+    let Z = {
+      let mut Z: Vec<Scalar> = (0..size_z)
+        .map(|_i| Scalar::random(&mut csprng))
+        .collect::<Vec<Scalar>>();
+      Z[num_vars] = Scalar::one(); // set the constant term to 1
+      Z
+    };
+
+    // three sparse matrices
+    let mut A: Vec<SparseMatEntry> = Vec::new();
+    let mut B: Vec<SparseMatEntry> = Vec::new();
+    let mut C: Vec<SparseMatEntry> = Vec::new();
+    let one = Scalar::one();
+    for i in 0..num_cons {
+      let A_idx = i % size_z;
+      let B_idx = (i + 2) % size_z;
+      A.push(SparseMatEntry::new(i, A_idx, one));
+      B.push(SparseMatEntry::new(i, B_idx, one));
+      let AB_val = Z[A_idx] * Z[B_idx];
+
+      let C_idx = (i + 3) % size_z;
+      let C_val = Z[C_idx];
+
+      if C_val == Scalar::zero() {
+        C.push(SparseMatEntry::new(i, num_vars, AB_val));
+      } else {
+        C.push(SparseMatEntry::new(
+          i,
+          C_idx,
+          AB_val * C_val.invert().unwrap(),
+        ));
+      }
+    }
+
+    let num_poly_vars_x = num_cons.log2();
+    let num_poly_vars_y = (2 * num_vars).log2();
+    let poly_A = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, A);
+    let poly_B = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, B);
+    let poly_C = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, C);
+
+    let inst = R1CSInstance::new(num_cons, num_vars, num_inputs, poly_A, poly_B, poly_C);
+
+    assert_eq!(
+      inst.is_sat(&Z[0..num_vars].to_vec(), &Z[num_vars + 1..].to_vec()),
+      true,
+    );
+
+    (inst, Z[0..num_vars].to_vec(), Z[num_vars + 1..].to_vec())
+  }
+
+  pub fn is_sat(&self, vars: &Vec<Scalar>, input: &Vec<Scalar>) -> bool {
+    assert_eq!(vars.len(), self.num_vars);
+    assert_eq!(input.len(), self.num_inputs);
+
+    let z = {
+      let mut z = vars.clone();
+      z.extend(&vec![Scalar::one()]);
+      z.extend(input);
+      z
+    };
+
+    // verify if Az * Bz - Cz = [0...]
+    let Az = self
+      .A
+      .multiply_vec(self.num_cons, self.num_vars + self.num_inputs + 1, &z);
+    let Bz = self
+      .B
+      .multiply_vec(self.num_cons, self.num_vars + self.num_inputs + 1, &z);
+    let Cz = self
+      .C
+      .multiply_vec(self.num_cons, self.num_vars + self.num_inputs + 1, &z);
+
+    assert_eq!(Az.len(), self.num_cons);
+    assert_eq!(Bz.len(), self.num_cons);
+    assert_eq!(Cz.len(), self.num_cons);
+    let res: usize = (0..self.num_cons)
+      .map(|i| if Az[i] * Bz[i] == Cz[i] { 0 } else { 1 })
+      .sum();
+    if res > 0 {
+      false
+    } else {
+      true
+    }
+  }
+
+  pub fn multiply_vec(
+    &self,
+    num_rows: usize,
+    num_cols: usize,
+    z: &Vec<Scalar>,
+  ) -> (DensePolynomial, DensePolynomial, DensePolynomial) {
+    assert_eq!(num_rows, self.num_cons);
+    assert_eq!(z.len(), num_cols);
+    assert!(num_cols > self.num_vars);
+    (
+      DensePolynomial::new(self.A.multiply_vec(num_rows, num_cols, z)),
+      DensePolynomial::new(self.B.multiply_vec(num_rows, num_cols, z)),
+      DensePolynomial::new(self.C.multiply_vec(num_rows, num_cols, z)),
+    )
+  }
+
+  pub fn compute_eval_table_sparse(
+    &self,
+    num_rows: usize,
+    num_cols: usize,
+    evals: &Vec<Scalar>,
+  ) -> (Vec<Scalar>, Vec<Scalar>, Vec<Scalar>) {
+    assert_eq!(num_rows, self.num_cons);
+    assert!(num_cols > self.num_vars);
+
+    let evals_A = self.A.compute_eval_table_sparse(&evals, num_rows, num_cols);
+    let evals_B = self.B.compute_eval_table_sparse(&evals, num_rows, num_cols);
+    let evals_C = self.C.compute_eval_table_sparse(&evals, num_rows, num_cols);
+
+    (evals_A, evals_B, evals_C)
+  }
+
+  pub fn evaluate_with_tables(
+    &self,
+    evals_rx: &Vec<Scalar>,
+    evals_ry: &Vec<Scalar>,
+  ) -> R1CSInstanceEvals {
+    R1CSInstanceEvals {
+      eval_A_r: self.A.evaluate_with_tables(evals_rx, evals_ry),
+      eval_B_r: self.B.evaluate_with_tables(evals_rx, evals_ry),
+      eval_C_r: self.C.evaluate_with_tables(evals_rx, evals_ry),
+    }
+  }
+
+  pub fn commit(&self, gens: &R1CSCommitmentGens) -> (R1CSCommitment, R1CSDecommitment) {
+    assert_eq!(self.A.get_num_nz_entries(), self.B.get_num_nz_entries());
+    assert_eq!(self.A.get_num_nz_entries(), self.C.get_num_nz_entries());
+    let (comm, dense) =
+      SparseMatPolynomial::multi_commit(&vec![&self.A, &self.B, &self.C], &gens.gens);
+    let r1cs_comm = R1CSCommitment {
+      num_cons: self.num_cons,
+      num_vars: self.num_vars,
+      num_inputs: self.num_inputs,
+      comm,
+    };
+
+    let r1cs_decomm = R1CSDecommitment { dense };
+
+    (r1cs_comm, r1cs_decomm)
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct R1CSEvalProof {
+  proof: SparseMatPolyEvalProof,
+}
+
+impl R1CSEvalProof {
+  pub fn prove(
+    decomm: &R1CSDecommitment,
+    rx: &Vec<Scalar>, // point at which the polynomial is evaluated
+    ry: &Vec<Scalar>,
+    evals: &R1CSInstanceEvals,
+    gens: &R1CSCommitmentGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+  ) -> R1CSEvalProof {
+    let timer = Timer::new("R1CSEvalProof::prove");
+    let proof = SparseMatPolyEvalProof::prove(
+      &decomm.dense,
+      rx,
+      ry,
+      &vec![evals.eval_A_r, evals.eval_B_r, evals.eval_C_r],
+      &gens.gens,
+      transcript,
+      random_tape,
+    );
+    timer.stop();
+
+    R1CSEvalProof { proof }
+  }
+
+  pub fn verify(
+    &self,
+    comm: &R1CSCommitment,
+    rx: &Vec<Scalar>, // point at which the R1CS matrix polynomials are evaluated
+    ry: &Vec<Scalar>,
+    eval: &R1CSInstanceEvals,
+    gens: &R1CSCommitmentGens,
+    transcript: &mut Transcript,
+  ) -> Result<(), ProofVerifyError> {
+    assert!(self
+      .proof
+      .verify(
+        &comm.comm,
+        rx,
+        ry,
+        &vec![eval.eval_A_r, eval.eval_B_r, eval.eval_C_r],
+        &gens.gens,
+        transcript
+      )
+      .is_ok());
+
+    Ok(())
+  }
+}

src/r1csproof.rs

@@ -0,0 +1,632 @@
+use super::commitments::{Commitments, MultiCommitGens};
+use super::dense_mlpoly::{
+  DensePolynomial, EqPolynomial, PolyCommitment, PolyCommitmentGens, PolyEvalProof,
+};
+use super::errors::ProofVerifyError;
+use super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
+use super::math::Math;
+use super::nizk::{EqualityProof, KnowledgeProof, ProductProof};
+use super::r1csinstance::{R1CSInstance, R1CSInstanceEvals};
+use super::scalar::Scalar;
+use super::sparse_mlpoly::{SparsePolyEntry, SparsePolynomial};
+use super::sumcheck::ZKSumcheckInstanceProof;
+use super::timer::Timer;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+use std::iter;
+
+#[cfg(test)]
+use super::sparse_mlpoly::{SparseMatEntry, SparseMatPolynomial};
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct R1CSProof {
+  comm_vars: PolyCommitment,
+  sc_proof_phase1: ZKSumcheckInstanceProof,
+  claims_phase2: (
+    CompressedGroup,
+    CompressedGroup,
+    CompressedGroup,
+    CompressedGroup,
+  ),
+  pok_claims_phase2: (KnowledgeProof, ProductProof),
+  proof_eq_sc_phase1: EqualityProof,
+  sc_proof_phase2: ZKSumcheckInstanceProof,
+  comm_vars_at_ry: CompressedGroup,
+  proof_eval_vars_at_ry: PolyEvalProof,
+  proof_eq_sc_phase2: EqualityProof,
+}
+
+pub struct R1CSSumcheckGens {
+  gens_1: MultiCommitGens,
+  gens_3: MultiCommitGens,
+  gens_4: MultiCommitGens,
+}
+
+// TODO: fix passing gens_1_ref
+impl R1CSSumcheckGens {
+  pub fn new(label: &'static [u8], gens_1_ref: &MultiCommitGens) -> Self {
+    let gens_1 = gens_1_ref.clone();
+    let gens_3 = MultiCommitGens::new(3, label);
+    let gens_4 = MultiCommitGens::new(4, label);
+
+    R1CSSumcheckGens {
+      gens_1,
+      gens_3,
+      gens_4,
+    }
+  }
+}
+
+pub struct R1CSGens {
+  gens_sc: R1CSSumcheckGens,
+  gens_pc: PolyCommitmentGens,
+}
+
+impl R1CSGens {
+  pub fn new(_num_cons: usize, num_vars: usize, label: &'static [u8]) -> Self {
+    let num_poly_vars = num_vars.log2();
+    let gens_pc = PolyCommitmentGens::new(num_poly_vars, label);
+    let gens_sc = R1CSSumcheckGens::new(label, &gens_pc.gens.gens_1);
+    R1CSGens { gens_sc, gens_pc }
+  }
+}
+
+impl R1CSProof {
+  fn prove_phase_one(
+    num_rounds: usize,
+    evals_tau: &mut DensePolynomial,
+    evals_Az: &mut DensePolynomial,
+    evals_Bz: &mut DensePolynomial,
+    evals_Cz: &mut DensePolynomial,
+    gens: &R1CSSumcheckGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+  ) -> (ZKSumcheckInstanceProof, Vec<Scalar>, Vec<Scalar>, Scalar) {
+    let comb_func = |poly_A_comp: &Scalar,
+                     poly_B_comp: &Scalar,
+                     poly_C_comp: &Scalar,
+                     poly_D_comp: &Scalar|
+     -> Scalar { poly_A_comp * (poly_B_comp * poly_C_comp - poly_D_comp) };
+
+    let (sc_proof_phase_one, r, claims, blind_claim_postsc) =
+      ZKSumcheckInstanceProof::prove_cubic_with_additive_term(
+        &Scalar::zero(), // claim is zero
+        &Scalar::zero(), // blind for claim is also zero
+        num_rounds,
+        evals_tau,
+        evals_Az,
+        evals_Bz,
+        evals_Cz,
+        comb_func,
+        &gens.gens_1,
+        &gens.gens_4,
+        transcript,
+        random_tape,
+      );
+
+    (sc_proof_phase_one, r, claims, blind_claim_postsc)
+  }
+
+  fn prove_phase_two(
+    num_rounds: usize,
+    claim: &Scalar,
+    blind_claim: &Scalar,
+    evals_z: &mut DensePolynomial,
+    evals_ABC: &mut DensePolynomial,
+    gens: &R1CSSumcheckGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+  ) -> (ZKSumcheckInstanceProof, Vec<Scalar>, Vec<Scalar>, Scalar) {
+    let comb_func =
+      |poly_A_comp: &Scalar, poly_B_comp: &Scalar| -> Scalar { poly_A_comp * poly_B_comp };
+    let (sc_proof_phase_two, r, claims, blind_claim_postsc) = ZKSumcheckInstanceProof::prove_quad(
+      claim,
+      blind_claim,
+      num_rounds,
+      evals_z,
+      evals_ABC,
+      comb_func,
+      &gens.gens_1,
+      &gens.gens_3,
+      transcript,
+      random_tape,
+    );
+
+    (sc_proof_phase_two, r, claims, blind_claim_postsc)
+  }
+
+  fn protocol_name() -> &'static [u8] {
+    b"R1CS proof"
+  }
+
+  pub fn prove(
+    inst: &R1CSInstance,
+    vars: Vec<Scalar>,
+    input: &Vec<Scalar>,
+    gens: &R1CSGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+  ) -> (R1CSProof, Vec<Scalar>, Vec<Scalar>) {
+    let timer_prove = Timer::new("R1CSProof::prove");
+    transcript.append_protocol_name(R1CSProof::protocol_name());
+
+    // we currently require the number of |inputs| + 1 to be at most number of vars
+    assert!(input.len() + 1 <= vars.len());
+
+    let timer_commit = Timer::new("polycommit");
+    let (poly_vars, comm_vars, blinds_vars) = {
+      // create a multilinear polynomial using the supplied assignment for variables
+      let poly_vars = DensePolynomial::new(vars.clone());
+
+      // produce a commitment to the satisfying assignment
+      let (comm_vars, blinds_vars) = poly_vars.commit(true, &gens.gens_pc, Some(random_tape));
+
+      // add the commitment to the prover's transcript
+      comm_vars.append_to_transcript(b"poly_commitment", transcript);
+      (poly_vars, comm_vars, blinds_vars)
+    };
+    timer_commit.stop();
+
+    let timer_sc_proof_phase1 = Timer::new("prove_sc_phase_one");
+
+    // append input to variables to create a single vector z
+    let z = {
+      let num_inputs = input.len();
+      let num_vars = vars.len();
+      let mut z = vars;
+      z.extend(&vec![Scalar::one()]); // add constant term in z
+      z.extend(input);
+      z.extend(&vec![Scalar::zero(); num_vars - num_inputs - 1]); // we will pad with zeros
+      z
+    };
+
+    // derive the verifier's challenge tau
+    let (num_rounds_x, num_rounds_y) = (inst.get_num_cons().log2(), z.len().log2());
+    let tau = transcript.challenge_vector(b"challenge_tau", num_rounds_x);
+    // compute the initial evaluation table for R(\tau, x)
+    let mut poly_tau = DensePolynomial::new(EqPolynomial::new(tau.clone()).evals());
+    let (mut poly_Az, mut poly_Bz, mut poly_Cz) =
+      inst.multiply_vec(inst.get_num_cons(), z.len(), &z);
+
+    let (sc_proof_phase1, rx, _claims_phase1, blind_claim_postsc1) = R1CSProof::prove_phase_one(
+      num_rounds_x,
+      &mut poly_tau,
+      &mut poly_Az,
+      &mut poly_Bz,
+      &mut poly_Cz,
+      &gens.gens_sc,
+      transcript,
+      random_tape,
+    );
+    assert_eq!(poly_tau.len(), 1);
+    assert_eq!(poly_Az.len(), 1);
+    assert_eq!(poly_Bz.len(), 1);
+    assert_eq!(poly_Cz.len(), 1);
+    timer_sc_proof_phase1.stop();
+
+    let (tau_claim, Az_claim, Bz_claim, Cz_claim) =
+      (&poly_tau[0], &poly_Az[0], &poly_Bz[0], &poly_Cz[0]);
+    let (Az_blind, Bz_blind, Cz_blind, prod_Az_Bz_blind) = (
+      random_tape.challenge_scalar(b"Az_blind"),
+      random_tape.challenge_scalar(b"Bz_blind"),
+      random_tape.challenge_scalar(b"Cz_blind"),
+      random_tape.challenge_scalar(b"prod_Az_Bz_blind"),
+    );
+
+    let (pok_Cz_claim, comm_Cz_claim) = {
+      KnowledgeProof::prove(
+        &gens.gens_sc.gens_1,
+        transcript,
+        random_tape,
+        &Cz_claim,
+        &Cz_blind,
+      )
+    };
+
+    let (proof_prod, comm_Az_claim, comm_Bz_claim, comm_prod_Az_Bz_claims) = {
+      let prod = Az_claim * Bz_claim;
+      ProductProof::prove(
+        &gens.gens_sc.gens_1,
+        transcript,
+        random_tape,
+        &Az_claim,
+        &Az_blind,
+        &Bz_claim,
+        &Bz_blind,
+        &prod,
+        &prod_Az_Bz_blind,
+      )
+    };
+
+    comm_Az_claim.append_to_transcript(b"comm_Az_claim", transcript);
+    comm_Bz_claim.append_to_transcript(b"comm_Bz_claim", transcript);
+    comm_Cz_claim.append_to_transcript(b"comm_Cz_claim", transcript);
+    comm_prod_Az_Bz_claims.append_to_transcript(b"comm_prod_Az_Bz_claims", transcript);
+
+    // prove the final step of sum-check #1
+    let taus_bound_rx = tau_claim;
+    let blind_expected_claim_postsc1 = taus_bound_rx * (&prod_Az_Bz_blind - &Cz_blind);
+    let claim_post_phase1 = (Az_claim * Bz_claim - Cz_claim) * taus_bound_rx;
+    let (proof_eq_sc_phase1, _C1, _C2) = EqualityProof::prove(
+      &gens.gens_sc.gens_1,
+      transcript,
+      random_tape,
+      &claim_post_phase1,
+      &blind_expected_claim_postsc1,
+      &claim_post_phase1,
+      &blind_claim_postsc1,
+    );
+
+    let timer_sc_proof_phase2 = Timer::new("prove_sc_phase_two");
+    // combine the three claims into a single claim
+    let r_A = transcript.challenge_scalar(b"challenege_Az");
+    let r_B = transcript.challenge_scalar(b"challenege_Bz");
+    let r_C = transcript.challenge_scalar(b"challenege_Cz");
+    let claim_phase2 = &r_A * Az_claim + &r_B * Bz_claim + &r_C * Cz_claim;
+    let blind_claim_phase2 = &r_A * Az_blind + &r_B * Bz_blind + &r_C * Cz_blind;
+
+    let evals_ABC = {
+      // compute the initial evaluation table for R(\tau, x)
+      let evals_rx = EqPolynomial::new(rx.clone()).evals();
+      let (evals_A, evals_B, evals_C) =
+        inst.compute_eval_table_sparse(inst.get_num_cons(), z.len(), &evals_rx);
+
+      assert_eq!(evals_A.len(), evals_B.len());
+      assert_eq!(evals_A.len(), evals_C.len());
+      (0..evals_A.len())
+        .map(|i| &r_A * &evals_A[i] + &r_B * &evals_B[i] + &r_C * &evals_C[i])
+        .collect::<Vec<Scalar>>()
+    };
+
+    // another instance of the sum-check protocol
+    let (sc_proof_phase2, ry, claims_phase2, blind_claim_postsc2) = R1CSProof::prove_phase_two(
+      num_rounds_y,
+      &claim_phase2,
+      &blind_claim_phase2,
+      &mut DensePolynomial::new(z),
+      &mut DensePolynomial::new(evals_ABC),
+      &gens.gens_sc,
+      transcript,
+      random_tape,
+    );
+    timer_sc_proof_phase2.stop();
+
+    let timer_polyeval = Timer::new("polyeval");
+    let eval_vars_at_ry = poly_vars.evaluate(&ry[1..].to_vec());
+    let blind_eval = random_tape.challenge_scalar(b"blind_eval");
+    let (proof_eval_vars_at_ry, comm_vars_at_ry) = PolyEvalProof::prove(
+      &poly_vars,
+      Some(&blinds_vars),
+      &ry[1..].to_vec(),
+      &eval_vars_at_ry,
+      Some(&blind_eval),
+      &gens.gens_pc,
+      transcript,
+      random_tape,
+    );
+    timer_polyeval.stop();
+
+    // prove the final step of sum-check #2
+    let blind_eval_Z_at_ry = (Scalar::one() - &ry[0]) * blind_eval;
+    let blind_expected_claim_postsc2 = &claims_phase2[1] * &blind_eval_Z_at_ry;
+    let claim_post_phase2 = &claims_phase2[0] * &claims_phase2[1];
+    let (proof_eq_sc_phase2, _C1, _C2) = EqualityProof::prove(
+      &gens.gens_pc.gens.gens_1,
+      transcript,
+      random_tape,
+      &claim_post_phase2,
+      &blind_expected_claim_postsc2,
+      &claim_post_phase2,
+      &blind_claim_postsc2,
+    );
+
+    timer_prove.stop();
+
+    (
+      R1CSProof {
+        comm_vars,
+        sc_proof_phase1,
+        claims_phase2: (
+          comm_Az_claim,
+          comm_Bz_claim,
+          comm_Cz_claim,
+          comm_prod_Az_Bz_claims,
+        ),
+        pok_claims_phase2: (pok_Cz_claim, proof_prod),
+        proof_eq_sc_phase1,
+        sc_proof_phase2,
+        comm_vars_at_ry,
+        proof_eval_vars_at_ry,
+        proof_eq_sc_phase2,
+      },
+      rx,
+      ry,
+    )
+  }
+
+  pub fn verify(
+    &self,
+    num_vars: usize,
+    num_cons: usize,
+    input: &Vec<Scalar>,
+    evals: &R1CSInstanceEvals,
+    transcript: &mut Transcript,
+    gens: &R1CSGens,
+  ) -> Result<(Vec<Scalar>, Vec<Scalar>), ProofVerifyError> {
+    transcript.append_protocol_name(R1CSProof::protocol_name());
+
+    let n = num_vars;
+    // add the commitment to the verifier's transcript
+    self
+      .comm_vars
+      .append_to_transcript(b"poly_commitment", transcript);
+
+    let (num_rounds_x, num_rounds_y) = (num_cons.log2(), (2 * num_vars).log2());
+
+    // derive the verifier's challenge tau
+    let tau = transcript.challenge_vector(b"challenge_tau", num_rounds_x);
+
+    // verify the first sum-check instance
+    let claim_phase1 = Scalar::zero()
+      .commit(&Scalar::zero(), &gens.gens_sc.gens_1)
+      .compress();
+    let (comm_claim_post_phase1, rx) = self
+      .sc_proof_phase1
+      .verify(
+        &claim_phase1,
+        num_rounds_x,
+        3,
+        &gens.gens_sc.gens_1,
+        &gens.gens_sc.gens_4,
+        transcript,
+      )
+      .unwrap();
+
+    // perform the intermediate sum-check test with claimed Az, Bz, and Cz
+    let (comm_Az_claim, comm_Bz_claim, comm_Cz_claim, comm_prod_Az_Bz_claims) = &self.claims_phase2;
+    let (pok_Cz_claim, proof_prod) = &self.pok_claims_phase2;
+
+    assert!(pok_Cz_claim
+      .verify(&gens.gens_sc.gens_1, transcript, &comm_Cz_claim)
+      .is_ok());
+    assert!(proof_prod
+      .verify(
+        &gens.gens_sc.gens_1,
+        transcript,
+        &comm_Az_claim,
+        &comm_Bz_claim,
+        &comm_prod_Az_Bz_claims
+      )
+      .is_ok());
+
+    comm_Az_claim.append_to_transcript(b"comm_Az_claim", transcript);
+    comm_Bz_claim.append_to_transcript(b"comm_Bz_claim", transcript);
+    comm_Cz_claim.append_to_transcript(b"comm_Cz_claim", transcript);
+    comm_prod_Az_Bz_claims.append_to_transcript(b"comm_prod_Az_Bz_claims", transcript);
+
+    let taus_bound_rx: Scalar = (0..rx.len())
+      .map(|i| &rx[i] * &tau[i] + (&Scalar::one() - &rx[i]) * (&Scalar::one() - &tau[i]))
+      .product();
+    let expected_claim_post_phase1 = (&taus_bound_rx
+      * (comm_prod_Az_Bz_claims.decompress().unwrap() - comm_Cz_claim.decompress().unwrap()))
+    .compress();
+
+    // verify proof that expected_claim_post_phase1 == claim_post_phase1
+    assert!(self
+      .proof_eq_sc_phase1
+      .verify(
+        &gens.gens_sc.gens_1,
+        transcript,
+        &expected_claim_post_phase1,
+        &comm_claim_post_phase1,
+      )
+      .is_ok());
+
+    // derive three public challenges and then derive a joint claim
+    let r_A = transcript.challenge_scalar(b"challenege_Az");
+    let r_B = transcript.challenge_scalar(b"challenege_Bz");
+    let r_C = transcript.challenge_scalar(b"challenege_Cz");
+
+    // r_A * comm_Az_claim + r_B * comm_Bz_claim + r_C * comm_Cz_claim;
+    let comm_claim_phase2 = GroupElement::vartime_multiscalar_mul(
+      iter::once(&r_A)
+        .chain(iter::once(&r_B))
+        .chain(iter::once(&r_C)),
+      iter::once(&comm_Az_claim)
+        .chain(iter::once(&comm_Bz_claim))
+        .chain(iter::once(&comm_Cz_claim))
+        .map(|pt| pt.decompress().unwrap())
+        .collect::<Vec<GroupElement>>(),
+    )
+    .compress();
+
+    // verify the joint claim with a sum-check protocol
+    let (comm_claim_post_phase2, ry) = self
+      .sc_proof_phase2
+      .verify(
+        &comm_claim_phase2,
+        num_rounds_y,
+        2,
+        &gens.gens_sc.gens_1,
+        &gens.gens_sc.gens_3,
+        transcript,
+      )
+      .unwrap();
+
+    // verify Z(ry) proof against the initial commitment
+    assert!(self
+      .proof_eval_vars_at_ry
+      .verify(
+        &gens.gens_pc,
+        transcript,
+        &ry[1..].to_vec(),
+        &self.comm_vars_at_ry,
+        &self.comm_vars
+      )
+      .is_ok());
+
+    let poly_input_eval = {
+      // constant term
+      let mut input_as_sparse_poly_entries = vec![SparsePolyEntry::new(0, Scalar::one())];
+      //remaining inputs
+      input_as_sparse_poly_entries.extend(
+        (0..input.len())
+          .map(|i| SparsePolyEntry::new(i + 1, input[i]))
+          .collect::<Vec<SparsePolyEntry>>(),
+      );
+      SparsePolynomial::new(n.log2(), input_as_sparse_poly_entries).evaluate(&ry[1..].to_vec())
+    };
+
+    // compute commitment to eval_Z_at_ry = (Scalar::one() - ry[0]) * self.eval_vars_at_ry + ry[0] * poly_input_eval
+    let comm_eval_Z_at_ry = GroupElement::vartime_multiscalar_mul(
+      iter::once(Scalar::one() - &ry[0]).chain(iter::once(ry[0])),
+      iter::once(&self.comm_vars_at_ry.decompress().unwrap()).chain(iter::once(
+        &poly_input_eval.commit(&Scalar::zero(), &gens.gens_pc.gens.gens_1),
+      )),
+    );
+
+    // perform the final check in the second sum-check protocol
+    let (eval_A_r, eval_B_r, eval_C_r) = evals.get_evaluations();
+    let expected_claim_post_phase2 =
+      (&(&r_A * &eval_A_r + &r_B * &eval_B_r + &r_C * &eval_C_r) * comm_eval_Z_at_ry).compress();
+    // verify proof that expected_claim_post_phase1 == claim_post_phase1
+    assert!(self
+      .proof_eq_sc_phase2
+      .verify(
+        &gens.gens_sc.gens_1,
+        transcript,
+        &expected_claim_post_phase2,
+        &comm_claim_post_phase2,
+      )
+      .is_ok());
+
+    Ok((rx, ry))
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+  use rand::rngs::OsRng;
+
+  fn produce_tiny_r1cs() -> (R1CSInstance, Vec<Scalar>, Vec<Scalar>) {
+    // three constraints over five variables Z1, Z2, Z3, Z4, and Z5
+    // rounded to the nearest power of two
+    let num_cons = 128;
+    let num_vars = 256;
+    let num_inputs = 2;
+
+    // encode the above constraints into three matrices
+    let mut A: Vec<SparseMatEntry> = Vec::new();
+    let mut B: Vec<SparseMatEntry> = Vec::new();
+    let mut C: Vec<SparseMatEntry> = Vec::new();
+
+    let one = Scalar::one();
+    // constraint 0 entries
+    // (Z1 + Z2) * I0 - Z3 = 0;
+    A.push(SparseMatEntry::new(0, 0, one));
+    A.push(SparseMatEntry::new(0, 1, one));
+    B.push(SparseMatEntry::new(0, num_vars + 1, one));
+    C.push(SparseMatEntry::new(0, 2, one));
+
+    // constraint 1 entries
+    // (Z1 + I1) * (Z3) - Z4 = 0
+    A.push(SparseMatEntry::new(1, 0, one));
+    A.push(SparseMatEntry::new(1, num_vars + 2, one));
+    B.push(SparseMatEntry::new(1, 2, one));
+    C.push(SparseMatEntry::new(1, 3, one));
+    // constraint 3 entries
+    // Z5 * 1 - 0 = 0
+    A.push(SparseMatEntry::new(2, 4, one));
+    B.push(SparseMatEntry::new(2, num_vars, one));
+
+    let num_vars_x = num_cons.log2();
+    let num_vars_y = (2 * num_vars).log2();
+
+    let poly_A = SparseMatPolynomial::new(num_vars_x, num_vars_y, A);
+    let poly_B = SparseMatPolynomial::new(num_vars_x, num_vars_y, B);
+    let poly_C = SparseMatPolynomial::new(num_vars_x, num_vars_y, C);
+
+    let inst = R1CSInstance::new(num_cons, num_vars, num_inputs, poly_A, poly_B, poly_C);
+
+    // compute a satisfying assignment
+    let mut csprng: OsRng = OsRng;
+    let i0 = Scalar::random(&mut csprng);
+    let i1 = Scalar::random(&mut csprng);
+    let z1 = Scalar::random(&mut csprng);
+    let z2 = Scalar::random(&mut csprng);
+    let z3 = (z1 + z2) * i0; // constraint 1: (Z1 + Z2) * I0 - Z3 = 0;
+    let z4 = (z1 + i1) * z3; // constraint 2: (Z1 + I1) * (Z3) - Z4 = 0
+    let z5 = Scalar::zero(); //constraint 3
+
+    let mut vars = vec![Scalar::zero(); num_vars];
+    vars[0] = z1;
+    vars[1] = z2;
+    vars[2] = z3;
+    vars[3] = z4;
+    vars[4] = z5;
+
+    let mut input = vec![Scalar::zero(); num_inputs];
+    input[0] = i0;
+    input[1] = i1;
+
+    (inst, vars, input)
+  }
+
+  #[test]
+  fn test_tiny_r1cs() {
+    let (inst, vars, input) = tests::produce_tiny_r1cs();
+    let is_sat = inst.is_sat(&vars, &input);
+    assert_eq!(is_sat, true);
+  }
+
+  #[test]
+  fn test_synthetic_r1cs() {
+    let (inst, vars, input) = R1CSInstance::produce_synthetic_r1cs(1024, 1024, 10);
+    let is_sat = inst.is_sat(&vars, &input);
+    assert_eq!(is_sat, true);
+  }
+
+  #[test]
+  pub fn check_r1cs_proof() {
+    let num_vars = 1024;
+    let num_cons = num_vars;
+    let num_inputs = 10;
+    let (inst, vars, input) = R1CSInstance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    let gens = R1CSGens::new(num_cons, num_vars, b"test-m");
+
+    let mut random_tape = {
+      let mut csprng: OsRng = OsRng;
+      let mut tape = Transcript::new(b"proof");
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut csprng));
+      tape
+    };
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, rx, ry) = R1CSProof::prove(
+      &inst,
+      vars,
+      &input,
+      &gens,
+      &mut prover_transcript,
+      &mut random_tape,
+    );
+
+    let eval_table_rx = EqPolynomial::new(rx).evals();
+    let eval_table_ry = EqPolynomial::new(ry).evals();
+    let inst_evals = inst.evaluate_with_tables(&eval_table_rx, &eval_table_ry);
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(
+        inst.get_num_vars(),
+        inst.get_num_cons(),
+        &input,
+        &inst_evals,
+        &mut verifier_transcript,
+        &gens,
+      )
+      .is_ok());
+  }
+}

src/scalar.rs

@@ -0,0 +1,48 @@
+pub type Scalar = super::scalar_25519::Scalar;
+pub type ScalarBytes = curve25519_dalek::scalar::Scalar;
+
+pub trait ScalarFromPrimitives {
+  fn to_scalar(self) -> Scalar;
+}
+
+impl ScalarFromPrimitives for usize {
+  #[inline]
+  fn to_scalar(self) -> Scalar {
+    (0..self).map(|_i| Scalar::one()).sum()
+  }
+}
+
+impl ScalarFromPrimitives for bool {
+  #[inline]
+  fn to_scalar(self) -> Scalar {
+    if self {
+      Scalar::one()
+    } else {
+      Scalar::zero()
+    }
+  }
+}
+
+pub trait ScalarBytesFromScalar {
+  fn decompress_scalar(s: &Scalar) -> ScalarBytes;
+  fn decompress_vec(v: &Vec<Scalar>) -> Vec<ScalarBytes>;
+  fn decompress_seq(s: &[Scalar]) -> Vec<ScalarBytes>;
+}
+
+impl ScalarBytesFromScalar for Scalar {
+  fn decompress_scalar(s: &Scalar) -> ScalarBytes {
+    ScalarBytes::from_bytes_mod_order(s.to_bytes())
+  }
+
+  fn decompress_vec(v: &Vec<Scalar>) -> Vec<ScalarBytes> {
+    (0..v.len())
+      .map(|i| Scalar::decompress_scalar(&v[i]))
+      .collect::<Vec<ScalarBytes>>()
+  }
+
+  fn decompress_seq(s: &[Scalar]) -> Vec<ScalarBytes> {
+    (0..s.len())
+      .map(|i| Scalar::decompress_scalar(&s[i]))
+      .collect::<Vec<ScalarBytes>>()
+  }
+}

src/spartan.rs

@@ -0,0 +1,190 @@
+use super::dense_mlpoly::EqPolynomial;
+use super::errors::ProofVerifyError;
+use super::r1csinstance::{
+  R1CSCommitment, R1CSCommitmentGens, R1CSDecommitment, R1CSEvalProof, R1CSInstance,
+  R1CSInstanceEvals,
+};
+use super::r1csproof::{R1CSGens, R1CSProof};
+use super::scalar::Scalar;
+use super::timer::Timer;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use merlin::Transcript;
+use rand::rngs::OsRng;
+use serde::{Deserialize, Serialize};
+
+pub struct SpartanGens {
+  gens_r1cs_sat: R1CSGens,
+  gens_r1cs_eval: R1CSCommitmentGens,
+}
+
+impl SpartanGens {
+  pub fn new(gens_r1cs_sat: R1CSGens, gens_r1cs_eval: R1CSCommitmentGens) -> SpartanGens {
+    SpartanGens {
+      gens_r1cs_sat,
+      gens_r1cs_eval,
+    }
+  }
+}
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct SpartanProof {
+  r1cs_sat_proof: R1CSProof,
+  inst_evals: R1CSInstanceEvals,
+  r1cs_eval_proof: R1CSEvalProof,
+}
+
+impl SpartanProof {
+  fn protocol_name() -> &'static [u8] {
+    b"Spartan proof"
+  }
+
+  /// A public computation to create a commitment to an R1CS instance
+  pub fn encode(
+    inst: &R1CSInstance,
+    gens: &R1CSCommitmentGens,
+  ) -> (R1CSCommitment, R1CSDecommitment) {
+    inst.commit(gens)
+  }
+
+  /// A method to produce a proof of the satisfiability of an R1CS instance
+  pub fn prove(
+    inst: &R1CSInstance,
+    decomm: &R1CSDecommitment,
+    vars: Vec<Scalar>,
+    input: &Vec<Scalar>,
+    gens: &SpartanGens,
+    transcript: &mut Transcript,
+  ) -> SpartanProof {
+    // we create a Transcript object seeded with a random Scalar
+    // to aid the prover produce its randomness
+    let mut random_tape = {
+      let mut csprng: OsRng = OsRng;
+      let mut tape = Transcript::new(b"SpartanProof");
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut csprng));
+      tape
+    };
+
+    transcript.append_protocol_name(SpartanProof::protocol_name());
+    let (r1cs_sat_proof, rx, ry) = {
+      let (proof, rx, ry) = R1CSProof::prove(
+        inst,
+        vars,
+        input,
+        &gens.gens_r1cs_sat,
+        transcript,
+        &mut random_tape,
+      );
+      let proof_encoded: Vec<u8> = bincode::serialize(&proof).unwrap();
+      Timer::print(&format!("len_r1cs_sat_proof {:?}", proof_encoded.len()));
+
+      (proof, rx, ry)
+    };
+
+    // We send evaluations of A, B, C at r = (rx, ry) as claims
+    // to enable the verifier complete the first sum-check
+    let timer_eval = Timer::new("eval_sparse_polys");
+    let inst_evals = {
+      let eval_table_rx = EqPolynomial::new(rx.clone()).evals();
+      let eval_table_ry = EqPolynomial::new(ry.clone()).evals();
+      inst.evaluate_with_tables(&eval_table_rx, &eval_table_ry)
+    };
+    inst_evals.append_to_transcript(b"r1cs_inst_evals", transcript);
+    timer_eval.stop();
+
+    let r1cs_eval_proof = {
+      let proof = R1CSEvalProof::prove(
+        decomm,
+        &rx,
+        &ry,
+        &inst_evals,
+        &gens.gens_r1cs_eval,
+        transcript,
+        &mut random_tape,
+      );
+
+      let proof_encoded: Vec<u8> = bincode::serialize(&proof).unwrap();
+      Timer::print(&format!("len_r1cs_eval_proof {:?}", proof_encoded.len()));
+      proof
+    };
+
+    SpartanProof {
+      r1cs_sat_proof,
+      inst_evals,
+      r1cs_eval_proof,
+    }
+  }
+
+  /// A method to verify the proof of the satisfiability of an R1CS instance
+  pub fn verify(
+    &self,
+    comm: &R1CSCommitment,
+    input: &Vec<Scalar>,
+    transcript: &mut Transcript,
+    gens: &SpartanGens,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(SpartanProof::protocol_name());
+
+    let timer_sat_proof = Timer::new("verify_sat_proof");
+    assert_eq!(input.len(), comm.get_num_inputs());
+    let (rx, ry) = self
+      .r1cs_sat_proof
+      .verify(
+        comm.get_num_vars(),
+        comm.get_num_cons(),
+        input,
+        &self.inst_evals,
+        transcript,
+        &gens.gens_r1cs_sat,
+      )
+      .unwrap();
+    timer_sat_proof.stop();
+
+    let timer_eval_proof = Timer::new("verify_eval_proof");
+    self
+      .inst_evals
+      .append_to_transcript(b"r1cs_inst_evals", transcript);
+    assert!(self
+      .r1cs_eval_proof
+      .verify(
+        comm,
+        &rx,
+        &ry,
+        &self.inst_evals,
+        &gens.gens_r1cs_eval,
+        transcript
+      )
+      .is_ok());
+    timer_eval_proof.stop();
+    Ok(())
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+
+  #[test]
+  pub fn check_spartan_proof() {
+    let num_vars = 256;
+    let num_cons = num_vars;
+    let num_inputs = 10;
+    let (inst, vars, input) = R1CSInstance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    let r1cs_size = inst.size();
+    let gens_r1cs_eval = R1CSCommitmentGens::new(&r1cs_size, b"gens_r1cs_eval");
+
+    // create a commitment to R1CSInstance
+    let (comm, decomm) = SpartanProof::encode(&inst, &gens_r1cs_eval);
+
+    let gens_r1cs_sat = R1CSGens::new(num_cons, num_vars, b"gens_r1cs_sat");
+    let gens = SpartanGens::new(gens_r1cs_sat, gens_r1cs_eval);
+
+    let mut prover_transcript = Transcript::new(b"example");
+    let proof = SpartanProof::prove(&inst, &decomm, vars, &input, &gens, &mut prover_transcript);
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&comm, &input, &mut verifier_transcript, &gens)
+      .is_ok());
+  }
+}

src/sumcheck.rs

@@ -0,0 +1,911 @@
+use super::commitments::{Commitments, MultiCommitGens};
+use super::dense_mlpoly::DensePolynomial;
+use super::errors::ProofVerifyError;
+use super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
+use super::nizk::DotProductProof;
+use super::scalar::Scalar;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use super::unipoly::{CompressedUniPoly, UniPoly};
+use itertools::izip;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+use std::iter;
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct SumcheckInstanceProof {
+  compressed_polys: Vec<CompressedUniPoly>,
+}
+
+impl SumcheckInstanceProof {
+  pub fn new(compressed_polys: Vec<CompressedUniPoly>) -> SumcheckInstanceProof {
+    SumcheckInstanceProof { compressed_polys }
+  }
+
+  pub fn verify(
+    &self,
+    claim: Scalar,
+    num_rounds: usize,
+    degree_bound: usize,
+    transcript: &mut Transcript,
+  ) -> Result<(Scalar, Vec<Scalar>), ProofVerifyError> {
+    let mut e = claim;
+    let mut r: Vec<Scalar> = Vec::new();
+
+    // verify that there is a univariate polynomial for each round
+    assert_eq!(self.compressed_polys.len(), num_rounds);
+    for i in 0..self.compressed_polys.len() {
+      let poly = self.compressed_polys[i].decompress(&e);
+
+      // verify degree bound
+      assert_eq!(poly.degree(), degree_bound);
+
+      // check if G_k(0) + G_k(1) = e
+      assert_eq!(poly.eval_at_zero() + poly.eval_at_one(), e);
+
+      // append the prover's message to the transcript
+      poly.append_to_transcript(b"poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_i = transcript.challenge_scalar(b"challenge_nextround");
+
+      r.push(r_i);
+
+      // evaluate the claimed degree-ell polynomial at r_i
+      e = poly.evaluate(&r_i);
+    }
+
+    Ok((e, r))
+  }
+}
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct ZKSumcheckInstanceProof {
+  comm_polys: Vec<CompressedGroup>,
+  comm_evals: Vec<CompressedGroup>,
+  proofs: Vec<DotProductProof>,
+}
+
+impl ZKSumcheckInstanceProof {
+  pub fn new(
+    comm_polys: Vec<CompressedGroup>,
+    comm_evals: Vec<CompressedGroup>,
+    proofs: Vec<DotProductProof>,
+  ) -> Self {
+    ZKSumcheckInstanceProof {
+      comm_polys,
+      comm_evals,
+      proofs,
+    }
+  }
+
+  pub fn verify(
+    &self,
+    comm_claim: &CompressedGroup,
+    num_rounds: usize,
+    degree_bound: usize,
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+  ) -> Result<(CompressedGroup, Vec<Scalar>), ProofVerifyError> {
+    // verify degree bound
+    assert_eq!(gens_n.n, degree_bound + 1);
+
+    // verify that there is a univariate polynomial for each round
+    assert_eq!(self.comm_polys.len(), num_rounds);
+    assert_eq!(self.comm_evals.len(), num_rounds);
+
+    let mut r: Vec<Scalar> = Vec::new();
+    for i in 0..self.comm_polys.len() {
+      let comm_poly = &self.comm_polys[i];
+
+      // append the prover's polynomial to the transcript
+      comm_poly.append_to_transcript(b"comm_poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_i = transcript.challenge_scalar(b"challenge_nextround");
+
+      // verify the proof of sum-check and evals
+      let res = {
+        let comm_claim_per_round = if i == 0 {
+          comm_claim
+        } else {
+          &self.comm_evals[i - 1]
+        };
+        let comm_eval = &self.comm_evals[i];
+
+        // add two claims to transcript
+        comm_claim_per_round.append_to_transcript(b"comm_claim_per_round", transcript);
+        comm_eval.append_to_transcript(b"comm_eval", transcript);
+
+        // produce two weights
+        let w = transcript.challenge_vector(b"combine_two_claims_to_one", 2);
+
+        // compute a weighted sum of the RHS
+        let comm_target = GroupElement::vartime_multiscalar_mul(
+          w.iter(),
+          iter::once(&comm_claim_per_round)
+            .chain(iter::once(&comm_eval))
+            .map(|pt| pt.decompress().unwrap())
+            .collect::<Vec<GroupElement>>(),
+        )
+        .compress();
+
+        let a = {
+          // the vector to use to decommit for sum-check test
+          let a_sc = {
+            let mut a = vec![Scalar::one(); degree_bound + 1];
+            a[0] = a[0] + Scalar::one();
+            a
+          };
+
+          // the vector to use to decommit for evaluation
+          let a_eval = {
+            let mut a = vec![Scalar::one(); degree_bound + 1];
+            for j in 1..a.len() {
+              a[j] = &a[j - 1] * &r_i;
+            }
+            a
+          };
+
+          // take weighted sum of the two vectors using w
+          assert_eq!(a_sc.len(), a_eval.len());
+          (0..a_sc.len())
+            .map(|i| &w[0] * &a_sc[i] + &w[1] * &a_eval[i])
+            .collect::<Vec<Scalar>>()
+        };
+
+        self.proofs[i]
+          .verify(
+            gens_1,
+            gens_n,
+            transcript,
+            &a,
+            &self.comm_polys[i],
+            &comm_target,
+          )
+          .is_ok()
+      };
+      assert!(res);
+
+      r.push(r_i);
+    }
+
+    Ok((self.comm_evals[self.comm_evals.len() - 1], r))
+  }
+}
+
+impl SumcheckInstanceProof {
+  pub fn prove_quad<F>(
+    claim: &Scalar,
+    num_rounds: usize,
+    poly_A: &mut DensePolynomial,
+    poly_B: &mut DensePolynomial,
+    comb_func: F,
+    transcript: &mut Transcript,
+  ) -> (Self, Vec<Scalar>, Vec<Scalar>)
+  where
+    F: Fn(&Scalar, &Scalar) -> Scalar,
+  {
+    let mut e = *claim;
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut quad_polys: Vec<CompressedUniPoly> = Vec::new();
+    for _j in 0..num_rounds {
+      let mut eval_point_0 = Scalar::zero();
+      let mut eval_point_2 = Scalar::zero();
+
+      let len = poly_A.len() / 2;
+      for i in 0..len {
+        // eval 0: bound_func is A(low)
+        eval_point_0 = &eval_point_0 + comb_func(&poly_A[i], &poly_B[i]);
+
+        // eval 2: bound_func is -A(low) + 2*A(high)
+        let poly_A_bound_point = &poly_A[len + i] + &poly_A[len + i] - &poly_A[i];
+        let poly_B_bound_point = &poly_B[len + i] + &poly_B[len + i] - &poly_B[i];
+        eval_point_2 = &eval_point_2 + comb_func(&poly_A_bound_point, &poly_B_bound_point);
+      }
+
+      let evals = vec![eval_point_0, e - eval_point_0, eval_point_2];
+      let poly = UniPoly::from_evals(&evals);
+
+      // append the prover's message to the transcript
+      poly.append_to_transcript(b"poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+      r.push(r_j);
+      // bound all tables to the verifier's challenege
+      poly_A.bound_poly_var_top(&r_j);
+      poly_B.bound_poly_var_top(&r_j);
+
+      e = poly.evaluate(&r_j);
+      quad_polys.push(poly.compress());
+    }
+
+    (
+      SumcheckInstanceProof::new(quad_polys),
+      r,
+      vec![poly_A[0], poly_B[0]],
+    )
+  }
+
+  pub fn prove_cubic<F>(
+    claim: &Scalar,
+    num_rounds: usize,
+    poly_A: &mut DensePolynomial,
+    poly_B: &mut DensePolynomial,
+    poly_C: &mut DensePolynomial,
+    comb_func: F,
+    transcript: &mut Transcript,
+  ) -> (Self, Vec<Scalar>, Vec<Scalar>)
+  where
+    F: Fn(&Scalar, &Scalar, &Scalar) -> Scalar,
+  {
+    let mut e = *claim;
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut cubic_polys: Vec<CompressedUniPoly> = Vec::new();
+    for _j in 0..num_rounds {
+      let mut eval_point_0 = Scalar::zero();
+      let mut eval_point_2 = Scalar::zero();
+      let mut eval_point_3 = Scalar::zero();
+
+      let len = poly_A.len() / 2;
+      for i in 0..len {
+        // eval 0: bound_func is A(low)
+        eval_point_0 = &eval_point_0 + comb_func(&poly_A[i], &poly_B[i], &poly_C[i]);
+
+        // eval 2: bound_func is -A(low) + 2*A(high)
+        let poly_A_bound_point = &poly_A[len + i] + &poly_A[len + i] - &poly_A[i];
+        let poly_B_bound_point = &poly_B[len + i] + &poly_B[len + i] - &poly_B[i];
+        let poly_C_bound_point = &poly_C[len + i] + &poly_C[len + i] - &poly_C[i];
+        eval_point_2 = &eval_point_2
+          + comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+          );
+
+        // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+        let poly_A_bound_point = &poly_A_bound_point + &poly_A[len + i] - &poly_A[i];
+        let poly_B_bound_point = &poly_B_bound_point + &poly_B[len + i] - &poly_B[i];
+        let poly_C_bound_point = &poly_C_bound_point + &poly_C[len + i] - &poly_C[i];
+
+        eval_point_3 = &eval_point_3
+          + comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+          );
+      }
+
+      let evals = vec![eval_point_0, e - eval_point_0, eval_point_2, eval_point_3];
+      let poly = UniPoly::from_evals(&evals);
+
+      // append the prover's message to the transcript
+      poly.append_to_transcript(b"poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+      r.push(r_j);
+      // bound all tables to the verifier's challenege
+      poly_A.bound_poly_var_top(&r_j);
+      poly_B.bound_poly_var_top(&r_j);
+      poly_C.bound_poly_var_top(&r_j);
+      e = poly.evaluate(&r_j);
+      cubic_polys.push(poly.compress());
+    }
+
+    (
+      SumcheckInstanceProof::new(cubic_polys),
+      r,
+      vec![poly_A[0], poly_B[0], poly_C[0]],
+    )
+  }
+
+  pub fn prove_cubic_with_additive_term<F>(
+    claim: &Scalar,
+    num_rounds: usize,
+    poly_A: &mut DensePolynomial,
+    poly_B: &mut DensePolynomial,
+    poly_C: &mut DensePolynomial,
+    poly_D: &mut DensePolynomial,
+    comb_func: F,
+    transcript: &mut Transcript,
+  ) -> (Self, Vec<Scalar>, Vec<Scalar>)
+  where
+    F: Fn(&Scalar, &Scalar, &Scalar, &Scalar) -> Scalar,
+  {
+    let mut e = *claim;
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut cubic_polys: Vec<CompressedUniPoly> = Vec::new();
+    for _j in 0..num_rounds {
+      let mut eval_point_0 = Scalar::zero();
+      let mut eval_point_2 = Scalar::zero();
+      let mut eval_point_3 = Scalar::zero();
+
+      let len = poly_A.len() / 2;
+      for i in 0..len {
+        // eval 0: bound_func is A(low)
+        eval_point_0 = &eval_point_0 + comb_func(&poly_A[i], &poly_B[i], &poly_C[i], &poly_D[i]);
+
+        // eval 2: bound_func is -A(low) + 2*A(high)
+        let poly_A_bound_point = &poly_A[len + i] + &poly_A[len + i] - &poly_A[i];
+        let poly_B_bound_point = &poly_B[len + i] + &poly_B[len + i] - &poly_B[i];
+        let poly_C_bound_point = &poly_C[len + i] + &poly_C[len + i] - &poly_C[i];
+        let poly_D_bound_point = &poly_D[len + i] + &poly_D[len + i] - &poly_D[i];
+        eval_point_2 = &eval_point_2
+          + comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+            &poly_D_bound_point,
+          );
+
+        // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+        let poly_A_bound_point = &poly_A_bound_point + &poly_A[len + i] - &poly_A[i];
+        let poly_B_bound_point = &poly_B_bound_point + &poly_B[len + i] - &poly_B[i];
+        let poly_C_bound_point = &poly_C_bound_point + &poly_C[len + i] - &poly_C[i];
+        let poly_D_bound_point = &poly_D_bound_point + &poly_D[len + i] - &poly_D[i];
+        eval_point_3 = &eval_point_3
+          + comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+            &poly_D_bound_point,
+          );
+      }
+
+      let evals = vec![eval_point_0, e - eval_point_0, eval_point_2, eval_point_3];
+      let poly = UniPoly::from_evals(&evals);
+
+      // append the prover's message to the transcript
+      poly.append_to_transcript(b"poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+      r.push(r_j);
+      // bound all tables to the verifier's challenege
+      poly_A.bound_poly_var_top(&r_j);
+      poly_B.bound_poly_var_top(&r_j);
+      poly_C.bound_poly_var_top(&r_j);
+      poly_D.bound_poly_var_top(&r_j);
+      e = poly.evaluate(&r_j);
+      cubic_polys.push(poly.compress());
+    }
+
+    (
+      SumcheckInstanceProof::new(cubic_polys),
+      r,
+      vec![poly_A[0], poly_B[0], poly_C[0], poly_D[0]],
+    )
+  }
+
+  pub fn prove_cubic_batched<F>(
+    claim: &Scalar,
+    num_rounds: usize,
+    poly_vec_par: (
+      &mut Vec<&mut DensePolynomial>,
+      &mut Vec<&mut DensePolynomial>,
+      &mut DensePolynomial,
+    ),
+    poly_vec_seq: (
+      &mut Vec<&mut DensePolynomial>,
+      &mut Vec<&mut DensePolynomial>,
+      &mut Vec<&mut DensePolynomial>,
+    ),
+    coeffs: &[Scalar],
+    comb_func: F,
+    transcript: &mut Transcript,
+  ) -> (
+    Self,
+    Vec<Scalar>,
+    (Vec<Scalar>, Vec<Scalar>, Scalar),
+    (Vec<Scalar>, Vec<Scalar>, Vec<Scalar>),
+  )
+  where
+    F: Fn(&Scalar, &Scalar, &Scalar) -> Scalar,
+  {
+    let (poly_A_vec_par, poly_B_vec_par, poly_C_par) = poly_vec_par;
+    let (poly_A_vec_seq, poly_B_vec_seq, poly_C_vec_seq) = poly_vec_seq;
+
+    //let (poly_A_vec_seq, poly_B_vec_seq, poly_C_vec_seq) = poly_vec_seq;
+    let mut e = *claim;
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut cubic_polys: Vec<CompressedUniPoly> = Vec::new();
+
+    for _j in 0..num_rounds {
+      let mut evals: Vec<(Scalar, Scalar, Scalar)> = Vec::new();
+
+      for (poly_A, poly_B) in poly_A_vec_par.iter().zip(poly_B_vec_par.iter()) {
+        let mut eval_point_0 = Scalar::zero();
+        let mut eval_point_2 = Scalar::zero();
+        let mut eval_point_3 = Scalar::zero();
+
+        let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 = &eval_point_0 + comb_func(&poly_A[i], &poly_B[i], &poly_C_par[i]);
+
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = &poly_A[len + i] + &poly_A[len + i] - &poly_A[i];
+          let poly_B_bound_point = &poly_B[len + i] + &poly_B[len + i] - &poly_B[i];
+          let poly_C_bound_point = &poly_C_par[len + i] + &poly_C_par[len + i] - &poly_C_par[i];
+          eval_point_2 = &eval_point_2
+            + comb_func(
+              &poly_A_bound_point,
+              &poly_B_bound_point,
+              &poly_C_bound_point,
+            );
+
+          // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+          let poly_A_bound_point = &poly_A_bound_point + &poly_A[len + i] - &poly_A[i];
+          let poly_B_bound_point = &poly_B_bound_point + &poly_B[len + i] - &poly_B[i];
+          let poly_C_bound_point = &poly_C_bound_point + &poly_C_par[len + i] - &poly_C_par[i];
+
+          eval_point_3 = &eval_point_3
+            + comb_func(
+              &poly_A_bound_point,
+              &poly_B_bound_point,
+              &poly_C_bound_point,
+            );
+        }
+
+        evals.push((eval_point_0, eval_point_2, eval_point_3));
+      }
+
+      for (poly_A, poly_B, poly_C) in izip!(
+        poly_A_vec_seq.iter(),
+        poly_B_vec_seq.iter(),
+        poly_C_vec_seq.iter()
+      ) {
+        let mut eval_point_0 = Scalar::zero();
+        let mut eval_point_2 = Scalar::zero();
+        let mut eval_point_3 = Scalar::zero();
+        let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 = &eval_point_0 + comb_func(&poly_A[i], &poly_B[i], &poly_C[i]);
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = &poly_A[len + i] + &poly_A[len + i] - &poly_A[i];
+          let poly_B_bound_point = &poly_B[len + i] + &poly_B[len + i] - &poly_B[i];
+          let poly_C_bound_point = &poly_C[len + i] + &poly_C[len + i] - &poly_C[i];
+          eval_point_2 = &eval_point_2
+            + comb_func(
+              &poly_A_bound_point,
+              &poly_B_bound_point,
+              &poly_C_bound_point,
+            );
+          // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+          let poly_A_bound_point = &poly_A_bound_point + &poly_A[len + i] - &poly_A[i];
+          let poly_B_bound_point = &poly_B_bound_point + &poly_B[len + i] - &poly_B[i];
+          let poly_C_bound_point = &poly_C_bound_point + &poly_C[len + i] - &poly_C[i];
+          eval_point_3 = &eval_point_3
+            + comb_func(
+              &poly_A_bound_point,
+              &poly_B_bound_point,
+              &poly_C_bound_point,
+            );
+        }
+        evals.push((eval_point_0, eval_point_2, eval_point_3));
+      }
+
+      let evals_combined_0 = (0..evals.len()).map(|i| evals[i].0 * coeffs[i]).sum();
+      let evals_combined_2 = (0..evals.len()).map(|i| evals[i].1 * coeffs[i]).sum();
+      let evals_combined_3 = (0..evals.len()).map(|i| evals[i].2 * coeffs[i]).sum();
+
+      let evals = vec![
+        evals_combined_0,
+        e - evals_combined_0,
+        evals_combined_2,
+        evals_combined_3,
+      ];
+      let poly = UniPoly::from_evals(&evals);
+
+      // append the prover's message to the transcript
+      poly.append_to_transcript(b"poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+      r.push(r_j);
+
+      // bound all tables to the verifier's challenege
+      for (poly_A, poly_B) in poly_A_vec_par.iter_mut().zip(poly_B_vec_par.iter_mut()) {
+        poly_A.bound_poly_var_top(&r_j);
+        poly_B.bound_poly_var_top(&r_j);
+      }
+      poly_C_par.bound_poly_var_top(&r_j);
+
+      for (poly_A, poly_B, poly_C) in izip!(
+        poly_A_vec_seq.iter_mut(),
+        poly_B_vec_seq.iter_mut(),
+        poly_C_vec_seq.iter_mut()
+      ) {
+        poly_A.bound_poly_var_top(&r_j);
+        poly_B.bound_poly_var_top(&r_j);
+        poly_C.bound_poly_var_top(&r_j);
+      }
+
+      e = poly.evaluate(&r_j);
+      cubic_polys.push(poly.compress());
+    }
+
+    let poly_A_par_final = (0..poly_A_vec_par.len())
+      .map(|i| poly_A_vec_par[i][0])
+      .collect();
+    let poly_B_par_final = (0..poly_B_vec_par.len())
+      .map(|i| poly_B_vec_par[i][0])
+      .collect();
+    let claims_prod = (poly_A_par_final, poly_B_par_final, poly_C_par[0]);
+
+    let poly_A_seq_final = (0..poly_A_vec_seq.len())
+      .map(|i| poly_A_vec_seq[i][0])
+      .collect();
+    let poly_B_seq_final = (0..poly_B_vec_seq.len())
+      .map(|i| poly_B_vec_seq[i][0])
+      .collect();
+    let poly_C_seq_final = (0..poly_C_vec_seq.len())
+      .map(|i| poly_C_vec_seq[i][0])
+      .collect();
+    let claims_dotp = (poly_A_seq_final, poly_B_seq_final, poly_C_seq_final);
+
+    (
+      SumcheckInstanceProof::new(cubic_polys),
+      r,
+      claims_prod,
+      claims_dotp,
+    )
+  }
+}
+
+impl ZKSumcheckInstanceProof {
+  pub fn prove_quad<F>(
+    claim: &Scalar,
+    blind_claim: &Scalar,
+    num_rounds: usize,
+    poly_A: &mut DensePolynomial,
+    poly_B: &mut DensePolynomial,
+    comb_func: F,
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+  ) -> (Self, Vec<Scalar>, Vec<Scalar>, Scalar)
+  where
+    F: Fn(&Scalar, &Scalar) -> Scalar,
+  {
+    let (blinds_poly, blinds_evals) = (
+      random_tape.challenge_vector(b"blinds_poly", num_rounds),
+      random_tape.challenge_vector(b"blinds_evals", num_rounds),
+    );
+    let mut claim_per_round = *claim;
+    let mut comm_claim_per_round = claim_per_round.commit(&blind_claim, &gens_1).compress();
+
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut comm_polys: Vec<CompressedGroup> = Vec::new();
+    let mut comm_evals: Vec<CompressedGroup> = Vec::new();
+    let mut proofs: Vec<DotProductProof> = Vec::new();
+
+    for j in 0..num_rounds {
+      let (poly, comm_poly) = {
+        let mut eval_point_0 = Scalar::zero();
+        let mut eval_point_2 = Scalar::zero();
+
+        let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 = &eval_point_0 + comb_func(&poly_A[i], &poly_B[i]);
+
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = &poly_A[len + i] + &poly_A[len + i] - &poly_A[i];
+          let poly_B_bound_point = &poly_B[len + i] + &poly_B[len + i] - &poly_B[i];
+          eval_point_2 = &eval_point_2 + comb_func(&poly_A_bound_point, &poly_B_bound_point);
+        }
+
+        let evals = vec![eval_point_0, claim_per_round - eval_point_0, eval_point_2];
+        let poly = UniPoly::from_evals(&evals);
+        let comm_poly = poly.commit(gens_n, &blinds_poly[j]).compress();
+        (poly, comm_poly)
+      };
+
+      // append the prover's message to the transcript
+      comm_poly.append_to_transcript(b"comm_poly", transcript);
+      comm_polys.push(comm_poly);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+
+      // bound all tables to the verifier's challenege
+      poly_A.bound_poly_var_top(&r_j);
+      poly_B.bound_poly_var_top(&r_j);
+
+      // produce a proof of sum-check and of evaluation
+      let (proof, claim_next_round, comm_claim_next_round) = {
+        let eval = poly.evaluate(&r_j);
+        let comm_eval = eval.commit(&blinds_evals[j], gens_1).compress();
+
+        // we need to prove the following under homomorphic commitments:
+        // (1) poly(0) + poly(1) = claim_per_round
+        // (2) poly(r_j) = eval
+
+        // Our technique is to leverage dot product proofs:
+        // (1) we can prove: <poly_in_coeffs_form, (2, 1, 1, 1)> = claim_per_round
+        // (2) we can prove: <poly_in_coeffs_form, (1, r_j, r^2_j, ..) = eval
+        // for efficiency we batch them using random weights
+
+        // add two claims to transcript
+        comm_claim_per_round.append_to_transcript(b"comm_claim_per_round", transcript);
+        comm_eval.append_to_transcript(b"comm_eval", transcript);
+
+        // produce two weights
+        let w = transcript.challenge_vector(b"combine_two_claims_to_one", 2);
+
+        // compute a weighted sum of the RHS
+        let target = &w[0] * &claim_per_round + &w[1] * &eval;
+        let comm_target = GroupElement::vartime_multiscalar_mul(
+          w.iter(),
+          iter::once(&comm_claim_per_round)
+            .chain(iter::once(&comm_eval))
+            .map(|pt| pt.decompress().unwrap())
+            .collect::<Vec<GroupElement>>(),
+        )
+        .compress();
+
+        let blind = {
+          let blind_sc = if j == 0 {
+            blind_claim
+          } else {
+            &blinds_evals[j - 1]
+          };
+
+          let blind_eval = &blinds_evals[j];
+
+          &w[0] * blind_sc + &w[1] * blind_eval
+        };
+        assert_eq!(target.commit(&blind, &gens_1).compress(), comm_target);
+
+        let a = {
+          // the vector to use to decommit for sum-check test
+          let a_sc = {
+            let mut a = vec![Scalar::one(); poly.degree() + 1];
+            a[0] = a[0] + Scalar::one();
+            a
+          };
+
+          // the vector to use to decommit for evaluation
+          let a_eval = {
+            let mut a = vec![Scalar::one(); poly.degree() + 1];
+            for j in 1..a.len() {
+              a[j] = &a[j - 1] * &r_j;
+            }
+            a
+          };
+
+          // take weighted sum of the two vectors using w
+          assert_eq!(a_sc.len(), a_eval.len());
+          (0..a_sc.len())
+            .map(|i| &w[0] * &a_sc[i] + &w[1] * &a_eval[i])
+            .collect::<Vec<Scalar>>()
+        };
+
+        let (proof, _comm_poly, _comm_sc_eval) = DotProductProof::prove(
+          gens_1,
+          gens_n,
+          transcript,
+          random_tape,
+          &poly.as_vec(),
+          &blinds_poly[j],
+          &a,
+          &target,
+          &blind,
+        );
+
+        (proof, eval, comm_eval)
+      };
+
+      claim_per_round = claim_next_round;
+      comm_claim_per_round = comm_claim_next_round;
+
+      proofs.push(proof);
+      r.push(r_j);
+      comm_evals.push(comm_claim_per_round);
+    }
+
+    (
+      ZKSumcheckInstanceProof::new(comm_polys, comm_evals, proofs),
+      r,
+      vec![poly_A[0], poly_B[0]],
+      blinds_evals[num_rounds - 1],
+    )
+  }
+
+  pub fn prove_cubic_with_additive_term<F>(
+    claim: &Scalar,
+    blind_claim: &Scalar,
+    num_rounds: usize,
+    poly_A: &mut DensePolynomial,
+    poly_B: &mut DensePolynomial,
+    poly_C: &mut DensePolynomial,
+    poly_D: &mut DensePolynomial,
+    comb_func: F,
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut Transcript,
+  ) -> (Self, Vec<Scalar>, Vec<Scalar>, Scalar)
+  where
+    F: Fn(&Scalar, &Scalar, &Scalar, &Scalar) -> Scalar,
+  {
+    let (blinds_poly, blinds_evals) = (
+      random_tape.challenge_vector(b"blinds_poly", num_rounds),
+      random_tape.challenge_vector(b"blinds_evals", num_rounds),
+    );
+
+    let mut claim_per_round = *claim;
+    let mut comm_claim_per_round = claim_per_round.commit(&blind_claim, &gens_1).compress();
+
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut comm_polys: Vec<CompressedGroup> = Vec::new();
+    let mut comm_evals: Vec<CompressedGroup> = Vec::new();
+    let mut proofs: Vec<DotProductProof> = Vec::new();
+
+    for j in 0..num_rounds {
+      let (poly, comm_poly) = {
+        let mut eval_point_0 = Scalar::zero();
+        let mut eval_point_2 = Scalar::zero();
+        let mut eval_point_3 = Scalar::zero();
+
+        let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 = &eval_point_0 + comb_func(&poly_A[i], &poly_B[i], &poly_C[i], &poly_D[i]);
+
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = &poly_A[len + i] + &poly_A[len + i] - &poly_A[i];
+          let poly_B_bound_point = &poly_B[len + i] + &poly_B[len + i] - &poly_B[i];
+          let poly_C_bound_point = &poly_C[len + i] + &poly_C[len + i] - &poly_C[i];
+          let poly_D_bound_point = &poly_D[len + i] + &poly_D[len + i] - &poly_D[i];
+          eval_point_2 = &eval_point_2
+            + comb_func(
+              &poly_A_bound_point,
+              &poly_B_bound_point,
+              &poly_C_bound_point,
+              &poly_D_bound_point,
+            );
+
+          // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+          let poly_A_bound_point = &poly_A_bound_point + &poly_A[len + i] - &poly_A[i];
+          let poly_B_bound_point = &poly_B_bound_point + &poly_B[len + i] - &poly_B[i];
+          let poly_C_bound_point = &poly_C_bound_point + &poly_C[len + i] - &poly_C[i];
+          let poly_D_bound_point = &poly_D_bound_point + &poly_D[len + i] - &poly_D[i];
+          eval_point_3 = &eval_point_3
+            + comb_func(
+              &poly_A_bound_point,
+              &poly_B_bound_point,
+              &poly_C_bound_point,
+              &poly_D_bound_point,
+            );
+        }
+
+        let evals = vec![
+          eval_point_0,
+          claim_per_round - eval_point_0,
+          eval_point_2,
+          eval_point_3,
+        ];
+        let poly = UniPoly::from_evals(&evals);
+        let comm_poly = poly.commit(gens_n, &blinds_poly[j]).compress();
+        (poly, comm_poly)
+      };
+
+      // append the prover's message to the transcript
+      comm_poly.append_to_transcript(b"comm_poly", transcript);
+      comm_polys.push(comm_poly);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+
+      // bound all tables to the verifier's challenege
+      poly_A.bound_poly_var_top(&r_j);
+      poly_B.bound_poly_var_top(&r_j);
+      poly_C.bound_poly_var_top(&r_j);
+      poly_D.bound_poly_var_top(&r_j);
+
+      // produce a proof of sum-check and of evaluation
+      let (proof, claim_next_round, comm_claim_next_round) = {
+        let eval = poly.evaluate(&r_j);
+        let comm_eval = eval.commit(&blinds_evals[j], gens_1).compress();
+
+        // we need to prove the following under homomorphic commitments:
+        // (1) poly(0) + poly(1) = claim_per_round
+        // (2) poly(r_j) = eval
+
+        // Our technique is to leverage dot product proofs:
+        // (1) we can prove: <poly_in_coeffs_form, (2, 1, 1, 1)> = claim_per_round
+        // (2) we can prove: <poly_in_coeffs_form, (1, r_j, r^2_j, ..) = eval
+        // for efficiency we batch them using random weights
+
+        // add two claims to transcript
+        comm_claim_per_round.append_to_transcript(b"comm_claim_per_round", transcript);
+        comm_eval.append_to_transcript(b"comm_eval", transcript);
+
+        // produce two weights
+        let w = transcript.challenge_vector(b"combine_two_claims_to_one", 2);
+
+        // compute a weighted sum of the RHS
+        let target = &w[0] * &claim_per_round + &w[1] * &eval;
+        let comm_target = GroupElement::vartime_multiscalar_mul(
+          w.iter(),
+          iter::once(&comm_claim_per_round)
+            .chain(iter::once(&comm_eval))
+            .map(|pt| pt.decompress().unwrap())
+            .collect::<Vec<GroupElement>>(),
+        )
+        .compress();
+
+        let blind = {
+          let blind_sc = if j == 0 {
+            blind_claim
+          } else {
+            &blinds_evals[j - 1]
+          };
+
+          let blind_eval = &blinds_evals[j];
+
+          &w[0] * blind_sc + &w[1] * blind_eval
+        };
+
+        assert_eq!(target.commit(&blind, &gens_1).compress(), comm_target);
+
+        let a = {
+          // the vector to use to decommit for sum-check test
+          let a_sc = {
+            let mut a = vec![Scalar::one(); poly.degree() + 1];
+            a[0] = a[0] + Scalar::one();
+            a
+          };
+
+          // the vector to use to decommit for evaluation
+          let a_eval = {
+            let mut a = vec![Scalar::one(); poly.degree() + 1];
+            for j in 1..a.len() {
+              a[j] = &a[j - 1] * &r_j;
+            }
+            a
+          };
+
+          // take weighted sum of the two vectors using w
+          assert_eq!(a_sc.len(), a_eval.len());
+          (0..a_sc.len())
+            .map(|i| &w[0] * &a_sc[i] + &w[1] * &a_eval[i])
+            .collect::<Vec<Scalar>>()
+        };
+
+        let (proof, _comm_poly, _comm_sc_eval) = DotProductProof::prove(
+          gens_1,
+          gens_n,
+          transcript,
+          random_tape,
+          &poly.as_vec(),
+          &blinds_poly[j],
+          &a,
+          &target,
+          &blind,
+        );
+
+        (proof, eval, comm_eval)
+      };
+
+      proofs.push(proof);
+      claim_per_round = claim_next_round;
+      comm_claim_per_round = comm_claim_next_round;
+      r.push(r_j);
+      comm_evals.push(comm_claim_per_round);
+    }
+
+    (
+      ZKSumcheckInstanceProof::new(comm_polys, comm_evals, proofs),
+      r,
+      vec![poly_A[0], poly_B[0], poly_C[0], poly_D[0]],
+      blinds_evals[num_rounds - 1],
+    )
+  }
+}

src/timer.rs

@@ -0,0 +1,83 @@
+#[cfg(feature = "profile")]
+use colored::Colorize;
+#[cfg(feature = "profile")]
+use std::sync::atomic::AtomicUsize;
+#[cfg(feature = "profile")]
+use std::{sync::atomic::Ordering, time::Instant};
+
+#[cfg(feature = "profile")]
+pub static CALL_DEPTH: AtomicUsize = AtomicUsize::new(0);
+
+#[cfg(feature = "profile")]
+pub struct Timer {
+  label: String,
+  timer: Instant,
+}
+
+#[cfg(feature = "profile")]
+impl Timer {
+  #[inline(always)]
+  pub fn new(label: &str) -> Self {
+    let timer = Instant::now();
+    CALL_DEPTH.fetch_add(1, Ordering::Relaxed);
+    println!(
+      "{:indent$}{}{}",
+      "",
+      "* ",
+      label.yellow().bold(),
+      indent = 2 * CALL_DEPTH.fetch_add(0, Ordering::Relaxed)
+    );
+    Self {
+      label: label.to_string(),
+      timer,
+    }
+  }
+
+  #[inline(always)]
+  pub fn stop(&self) {
+    let duration = self.timer.elapsed();
+    println!(
+      "{:indent$}{}{} {:?}",
+      "",
+      "* ",
+      self.label.blue().bold(),
+      duration,
+      indent = 2 * CALL_DEPTH.fetch_add(0, Ordering::Relaxed)
+    );
+    CALL_DEPTH.fetch_sub(1, Ordering::Relaxed);
+  }
+
+  #[inline(always)]
+  pub fn print(msg: &str) {
+    CALL_DEPTH.fetch_add(1, Ordering::Relaxed);
+    println!(
+      "{:indent$}{}{}",
+      "",
+      "* ",
+      msg.to_string().green().bold(),
+      indent = 2 * CALL_DEPTH.fetch_add(0, Ordering::Relaxed)
+    );
+    CALL_DEPTH.fetch_sub(1, Ordering::Relaxed);
+  }
+}
+
+#[cfg(not(feature = "profile"))]
+pub struct Timer {
+  _label: String,
+}
+
+#[cfg(not(feature = "profile"))]
+impl Timer {
+  #[inline(always)]
+  pub fn new(label: &str) -> Self {
+    Self {
+      _label: label.to_string(),
+    }
+  }
+
+  #[inline(always)]
+  pub fn stop(&self) {}
+
+  #[inline(always)]
+  pub fn print(_msg: &str) {}
+}

src/transcript.rs

@@ -0,0 +1,63 @@
+use super::group::CompressedGroup;
+use super::scalar::Scalar;
+use merlin::Transcript;
+
+pub trait ProofTranscript {
+  fn append_protocol_name(&mut self, protocol_name: &'static [u8]);
+  fn append_scalar(&mut self, label: &'static [u8], scalar: &Scalar);
+  fn append_point(&mut self, label: &'static [u8], point: &CompressedGroup);
+  fn challenge_scalar(&mut self, label: &'static [u8]) -> Scalar;
+  fn challenge_vector(&mut self, label: &'static [u8], len: usize) -> Vec<Scalar>;
+}
+
+impl ProofTranscript for Transcript {
+  fn append_protocol_name(&mut self, protocol_name: &'static [u8]) {
+    self.append_message(b"protocol-name", protocol_name);
+  }
+
+  fn append_scalar(&mut self, label: &'static [u8], scalar: &Scalar) {
+    self.append_message(label, &scalar.to_bytes());
+  }
+
+  fn append_point(&mut self, label: &'static [u8], point: &CompressedGroup) {
+    self.append_message(label, point.as_bytes());
+  }
+
+  fn challenge_scalar(&mut self, label: &'static [u8]) -> Scalar {
+    let mut buf = [0u8; 64];
+    self.challenge_bytes(label, &mut buf);
+    Scalar::from_bytes_wide(&buf)
+  }
+
+  fn challenge_vector(&mut self, label: &'static [u8], len: usize) -> Vec<Scalar> {
+    (0..len)
+      .map(|_i| self.challenge_scalar(label))
+      .collect::<Vec<Scalar>>()
+  }
+}
+
+pub trait AppendToTranscript {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript);
+}
+
+impl AppendToTranscript for Scalar {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_scalar(label, self);
+  }
+}
+
+impl AppendToTranscript for Vec<Scalar> {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_message(label, b"begin_append_vector");
+    for i in 0..self.len() {
+      transcript.append_scalar(label, &self[i]);
+    }
+    transcript.append_message(label, b"end_append_vector");
+  }
+}
+
+impl AppendToTranscript for CompressedGroup {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_point(label, self);
+  }
+}

src/unipoly.rs

@@ -0,0 +1,184 @@
+use super::commitments::{Commitments, MultiCommitGens};
+use super::group::GroupElement;
+use super::scalar::{Scalar, ScalarFromPrimitives};
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+// ax^2 + bx + c stored as vec![a,b,c]
+// ax^3 + bx^2 + cx + d stored as vec![a,b,c,d]
+#[derive(Debug)]
+pub struct UniPoly {
+  coeffs: Vec<Scalar>,
+}
+
+// ax^2 + bx + c stored as vec![a,c]
+// ax^3 + bx^2 + cx + d stored as vec![a,c,d]
+#[derive(Serialize, Deserialize, Debug)]
+pub struct CompressedUniPoly {
+  coeffs_except_linear_term: Vec<Scalar>,
+}
+
+impl UniPoly {
+  pub fn from_evals(evals: &Vec<Scalar>) -> Self {
+    // we only support degree-2 or degree-3 univariate polynomials
+    assert!(evals.len() == 3 || evals.len() == 4);
+    let coeffs = if evals.len() == 3 {
+      // ax^2 + bx + c
+      let two_inv = (2 as usize).to_scalar().invert().unwrap();
+
+      let c = evals[0];
+      let a = two_inv * (evals[2] - evals[1] - evals[1] + c);
+      let b = evals[1] - c - a;
+      vec![c, b, a]
+    } else {
+      // ax^3 + bx^2 + cx + d
+      let two_inv = (2 as usize).to_scalar().invert().unwrap();
+      let six_inv = (6 as usize).to_scalar().invert().unwrap();
+
+      let d = evals[0];
+      let a = six_inv
+        * (evals[3] - evals[2] - evals[2] - evals[2] + evals[1] + evals[1] + evals[1] - evals[0]);
+      let b = two_inv
+        * (evals[0] + evals[0] - evals[1] - evals[1] - evals[1] - evals[1] - evals[1]
+          + evals[2]
+          + evals[2]
+          + evals[2]
+          + evals[2]
+          - evals[3]);
+      let c = evals[1] - d - a - b;
+      vec![d, c, b, a]
+    };
+
+    UniPoly { coeffs }
+  }
+
+  pub fn degree(&self) -> usize {
+    self.coeffs.len() - 1
+  }
+
+  pub fn as_vec(&self) -> Vec<Scalar> {
+    self.coeffs.clone()
+  }
+
+  pub fn eval_at_zero(&self) -> Scalar {
+    self.coeffs[0]
+  }
+
+  pub fn eval_at_one(&self) -> Scalar {
+    (0..self.coeffs.len()).map(|i| self.coeffs[i]).sum()
+  }
+
+  pub fn evaluate(&self, r: &Scalar) -> Scalar {
+    let mut eval = self.coeffs[0];
+    let mut power = *r;
+    for i in 1..self.coeffs.len() {
+      eval = &eval + &power * &self.coeffs[i];
+      power = &power * r;
+    }
+    eval
+  }
+
+  pub fn compress(&self) -> CompressedUniPoly {
+    let coeffs_except_linear_term = [&self.coeffs[0..1], &self.coeffs[2..]].concat();
+    assert_eq!(coeffs_except_linear_term.len() + 1, self.coeffs.len());
+    CompressedUniPoly {
+      coeffs_except_linear_term,
+    }
+  }
+
+  pub fn commit(&self, gens: &MultiCommitGens, blind: &Scalar) -> GroupElement {
+    self.coeffs.commit(blind, gens)
+  }
+}
+
+impl CompressedUniPoly {
+  // we require eval(0) + eval(1) = hint, so we can solve for the linear term as:
+  // linear_term = hint - 2 * constant_term - deg2 term - deg3 term
+  pub fn decompress(&self, hint: &Scalar) -> UniPoly {
+    let mut linear_term =
+      hint - self.coeffs_except_linear_term[0] - self.coeffs_except_linear_term[0];
+    for i in 1..self.coeffs_except_linear_term.len() {
+      linear_term = linear_term - self.coeffs_except_linear_term[i];
+    }
+
+    let mut coeffs: Vec<Scalar> = Vec::new();
+    coeffs.extend(vec![&self.coeffs_except_linear_term[0]]);
+    coeffs.extend(vec![&linear_term]);
+    coeffs.extend(self.coeffs_except_linear_term[1..].to_vec());
+    assert_eq!(self.coeffs_except_linear_term.len() + 1, coeffs.len());
+    UniPoly { coeffs }
+  }
+}
+
+impl AppendToTranscript for UniPoly {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_message(label, b"UniPoly_begin");
+    for i in 0..self.coeffs.len() {
+      transcript.append_scalar(b"coeff", &self.coeffs[i]);
+    }
+    transcript.append_message(label, b"UniPoly_end");
+  }
+}
+
+#[cfg(test)]
+mod tests {
+
+  use super::*;
+
+  #[test]
+  fn test_from_evals_quad() {
+    // polynomial is 2x^2 + 3x + 1
+    let e0 = Scalar::one();
+    let e1 = (6 as usize).to_scalar();
+    let e2 = (15 as usize).to_scalar();
+    let evals = vec![e0, e1, e2];
+    let poly = UniPoly::from_evals(&evals);
+
+    assert_eq!(poly.eval_at_zero(), e0);
+    assert_eq!(poly.eval_at_one(), e1);
+    assert_eq!(poly.coeffs.len(), 3);
+    assert_eq!(poly.coeffs[0], Scalar::one());
+    assert_eq!(poly.coeffs[1], (3 as usize).to_scalar());
+    assert_eq!(poly.coeffs[2], (2 as usize).to_scalar());
+
+    let hint = e0 + e1;
+    let compressed_poly = poly.compress();
+    let decompressed_poly = compressed_poly.decompress(&hint);
+    for i in 0..decompressed_poly.coeffs.len() {
+      assert_eq!(decompressed_poly.coeffs[i], poly.coeffs[i]);
+    }
+
+    let e3 = (28 as usize).to_scalar();
+    assert_eq!(poly.evaluate(&(3 as usize).to_scalar()), e3);
+  }
+
+  #[test]
+  fn test_from_evals_cubic() {
+    // polynomial is x^3 + 2x^2 + 3x + 1
+    let e0 = Scalar::one();
+    let e1 = (7 as usize).to_scalar();
+    let e2 = (23 as usize).to_scalar();
+    let e3 = (55 as usize).to_scalar();
+    let evals = vec![e0, e1, e2, e3];
+    let poly = UniPoly::from_evals(&evals);
+
+    assert_eq!(poly.eval_at_zero(), e0);
+    assert_eq!(poly.eval_at_one(), e1);
+    assert_eq!(poly.coeffs.len(), 4);
+    assert_eq!(poly.coeffs[0], Scalar::one());
+    assert_eq!(poly.coeffs[1], (3 as usize).to_scalar());
+    assert_eq!(poly.coeffs[2], (2 as usize).to_scalar());
+    assert_eq!(poly.coeffs[3], (1 as usize).to_scalar());
+
+    let hint = e0 + e1;
+    let compressed_poly = poly.compress();
+    let decompressed_poly = compressed_poly.decompress(&hint);
+    for i in 0..decompressed_poly.coeffs.len() {
+      assert_eq!(decompressed_poly.coeffs[i], poly.coeffs[i]);
+    }
+
+    let e4 = (109 as usize).to_scalar();
+    assert_eq!(poly.evaluate(&(4 as usize).to_scalar()), e4);
+  }
+}