@@ -11,7 +11,6 @@ license-file = "LICENSE"
 keywords = ["Recursive zkSNARKs", "cryptography", "proofs"]
 
 [dependencies]
-curve25519-dalek = {version = "3.0.0", features = ["simd_backend"]}
 merlin = "2.0.0"
 rand = "0.7.3"
 digest = "0.8.1"
@@ -19,3 +18,7 @@ sha3 = "0.8.2"
 rayon = "1.3.0"
 rand_core = { version = "0.5", default-features = false }
 itertools = "0.9.0"
+subtle = "2.4"
+
+[dev-dependencies]
+curve25519-dalek = {version = "3.0.0", features = ["simd_backend"]}

@@ -1,56 +1,53 @@
 use super::errors::NovaError;
+use super::traits::{CompressedGroup, Group};
+use core::fmt::Debug;
 use core::ops::{Add, AddAssign, Mul, MulAssign};
-use curve25519_dalek::traits::VartimeMultiscalarMul;
 use digest::{ExtendableOutput, Input};
 use merlin::Transcript;
 use sha3::Shake256;
 use std::io::Read;
-
-pub type Scalar = curve25519_dalek::scalar::Scalar;
-type GroupElement = curve25519_dalek::ristretto::RistrettoPoint;
-type CompressedGroup = curve25519_dalek::ristretto::CompressedRistretto;
 
 #[derive(Debug)]
-pub struct CommitGens {
-  gens: Vec<GroupElement>,
+pub struct CommitGens<G: Group> {
+  gens: Vec<G>,
 }
 
-#[derive(Clone, Debug, PartialEq, Eq)]
-pub struct Commitment {
-  comm: GroupElement,
+#[derive(Clone, Copy, Debug, PartialEq, Eq)]
+pub struct Commitment<G: Group> {
+  comm: G,
 }
 
 #[derive(Clone, Debug, PartialEq, Eq)]
-pub struct CompressedCommitment {
-  comm: CompressedGroup,
+pub struct CompressedCommitment<C: CompressedGroup> {
+  comm: C,
 }
 
-impl CommitGens {
+impl<G: Group> CommitGens<G> {
   pub fn new(label: &[u8], n: usize) -> Self {
     let mut shake = Shake256::default();
     shake.input(label);
     let mut reader = shake.xof_result();
-    let mut gens: Vec<GroupElement> = Vec::new();
+    let mut gens: Vec<G> = Vec::new();
     let mut uniform_bytes = [0u8; 64];
     for _ in 0..n {
       reader.read_exact(&mut uniform_bytes).unwrap();
-      gens.push(GroupElement::from_uniform_bytes(&uniform_bytes));
+      gens.push(G::from_uniform_bytes(&uniform_bytes).unwrap());
     }
 
     CommitGens { gens }
   }
 }
 
-impl Commitment {
-  pub fn compress(&self) -> CompressedCommitment {
+impl<G: Group> Commitment<G> {
+  pub fn compress(&self) -> CompressedCommitment<G::CompressedGroupElement> {
     CompressedCommitment {
       comm: self.comm.compress(),
     }
   }
 }
 
-impl CompressedCommitment {
-  pub fn decompress(&self) -> Result<Commitment, NovaError> {
+impl<C: CompressedGroup> CompressedCommitment<C> {
+  pub fn decompress(&self) -> Result<Commitment<C::GroupElement>, NovaError> {
     let comm = self.comm.decompress();
     if comm.is_none() {
       return Err(NovaError::DecompressionError);
@@ -61,161 +58,90 @@ })
   }
 }
 
-pub trait CommitTrait {
-  fn commit(&self, gens: &CommitGens) -> Commitment;
+pub trait CommitTrait<G: Group> {
+  fn commit(&self, gens: &CommitGens<G>) -> Commitment<G>;
 }
 
-impl CommitTrait for [Scalar] {
-  fn commit(&self, gens: &CommitGens) -> Commitment {
+impl<G: Group> CommitTrait<G> for [G::Scalar] {
+  fn commit(&self, gens: &CommitGens<G>) -> Commitment<G> {
     assert_eq!(gens.gens.len(), self.len());
     Commitment {
-      comm: GroupElement::vartime_multiscalar_mul(self, &gens.gens),
+      comm: G::vartime_multiscalar_mul(self, &gens.gens),
     }
   }
 }
 
-pub trait ProofTranscriptTrait {
-  fn append_protocol_name(&mut self, protocol_name: &'static [u8]);
-  fn challenge_scalar(&mut self, label: &'static [u8]) -> Scalar;
+pub trait AppendToTranscriptTrait {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript);
 }
 
-impl ProofTranscriptTrait for Transcript {
-  fn append_protocol_name(&mut self, protocol_name: &'static [u8]) {
-    self.append_message(b"protocol-name", protocol_name);
-  }
-
-  fn challenge_scalar(&mut self, label: &'static [u8]) -> Scalar {
-    let mut buf = [0u8; 64];
-    self.challenge_bytes(label, &mut buf);
-    Scalar::from_bytes_mod_order_wide(&buf)
+impl<G: Group> AppendToTranscriptTrait for Commitment<G> {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_message(label, self.comm.compress().as_bytes());
   }
 }
 
-pub trait AppendToTranscriptTrait {
-  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript);
-}
-
-impl AppendToTranscriptTrait for CompressedCommitment {
+impl<C: CompressedGroup> AppendToTranscriptTrait for CompressedCommitment<C> {
   fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
     transcript.append_message(label, self.comm.as_bytes());
   }
 }
 
-impl<'b> MulAssign<&'b Scalar> for Commitment {
-  fn mul_assign(&mut self, scalar: &'b Scalar) {
-    let result = (self as &Commitment).comm * scalar;
+impl<'b, G: Group> MulAssign<&'b G::Scalar> for Commitment<G> {
+  fn mul_assign(&mut self, scalar: &'b G::Scalar) {
+    let result = (self as &Commitment<G>).comm * scalar;
     *self = Commitment { comm: result };
   }
 }
 
-impl<'a, 'b> Mul<&'b Scalar> for &'a Commitment {
-  type Output = Commitment;
-  fn mul(self, scalar: &'b Scalar) -> Commitment {
+impl<G: Group> Mul<G::Scalar> for Commitment<G> {
+  type Output = Commitment<G>;
+  fn mul(self, scalar: G::Scalar) -> Commitment<G> {
     Commitment {
       comm: self.comm * scalar,
     }
   }
 }
 
-impl<'a, 'b> Mul<&'b Commitment> for &'a Scalar {
-  type Output = Commitment;
-
-  fn mul(self, comm: &'b Commitment) -> Commitment {
-    Commitment {
-      comm: self * comm.comm,
-    }
-  }
-}
-
-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 = Commitment, RHS = Scalar);
-define_mul_variants!(LHS = Commitment, RHS = Scalar, Output = Commitment);
-define_mul_variants!(LHS = Scalar, RHS = Commitment, Output = Commitment);
-
-impl<'b> AddAssign<&'b Commitment> for Commitment {
-  fn add_assign(&mut self, other: &'b Commitment) {
-    let result = (self as &Commitment).comm + other.comm;
+impl<'b, G: Group> AddAssign<&'b Commitment<G>> for Commitment<G> {
+  fn add_assign(&mut self, other: &'b Commitment<G>) {
+    let result = (self as &Commitment<G>).comm + other.comm;
     *self = Commitment { comm: result };
   }
 }
 
-impl<'a, 'b> Add<&'b Commitment> for &'a Commitment {
-  type Output = Commitment;
-  fn add(self, other: &'b Commitment) -> Commitment {
+impl<'a, 'b, G: Group> Add<&'b Commitment<G>> for &'a Commitment<G> {
+  type Output = Commitment<G>;
+  fn add(self, other: &'b Commitment<G>) -> Commitment<G> {
     Commitment {
       comm: self.comm + other.comm,
     }
   }
 }
 
-macro_rules! define_add_variants {
-  (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {
-    impl<'b> Add<&'b $rhs> for $lhs {
-      type Output = $out;
-      fn add(self, rhs: &'b $rhs) -> $out {
-        &self + rhs
-      }
-    }
+impl<G: Group> AddAssign<Commitment<G>> for Commitment<G> {
+  fn add_assign(&mut self, rhs: Commitment<G>) {
+    *self += &rhs;
+  }
+}
 
-    impl<'a> Add<$rhs> for &'a $lhs {
-      type Output = $out;
-      fn add(self, rhs: $rhs) -> $out {
-        self + &rhs
-      }
-    }
-
-    impl Add<$rhs> for $lhs {
-      type Output = $out;
-      fn add(self, rhs: $rhs) -> $out {
-        &self + &rhs
-      }
-    }
-  };
+impl<'b, G: Group> Add<&'b Commitment<G>> for Commitment<G> {
+  type Output = Commitment<G>;
+  fn add(self, rhs: &'b Commitment<G>) -> Commitment<G> {
+    &self + rhs
+  }
 }
 
-macro_rules! define_add_assign_variants {
-  (LHS = $lhs:ty, RHS = $rhs:ty) => {
-    impl AddAssign<$rhs> for $lhs {
-      fn add_assign(&mut self, rhs: $rhs) {
-        *self += &rhs;
-      }
-    }
-  };
+impl<'a, G: Group> Add<Commitment<G>> for &'a Commitment<G> {
+  type Output = Commitment<G>;
+  fn add(self, rhs: Commitment<G>) -> Commitment<G> {
+    self + &rhs
+  }
 }
 
-define_add_assign_variants!(LHS = Commitment, RHS = Commitment);
-define_add_variants!(LHS = Commitment, RHS = Commitment, Output = Commitment);
+impl<G: Group> Add<Commitment<G>> for Commitment<G> {
+  type Output = Commitment<G>;
+  fn add(self, rhs: Commitment<G>) -> Commitment<G> {
+    &self + &rhs
+  }
+}

@@ -1,33 +1,25 @@
+//! This library implements core components of Nova.
 #![allow(non_snake_case)]
 #![feature(test)]
 #![deny(missing_docs)]
-#![feature(external_doc)]
-#![doc(include = "../README.md")]
-
-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 commitments;
 mod errors;
 mod r1cs;
+mod traits;
 
-use commitments::{AppendToTranscriptTrait, CompressedCommitment, ProofTranscriptTrait};
+use commitments::{AppendToTranscriptTrait, Commitment};
 use errors::NovaError;
 use merlin::Transcript;
 use r1cs::{R1CSGens, R1CSInstance, R1CSShape, R1CSWitness};
+use traits::{ChallengeTrait, Group};
 
 /// A SNARK that holds the proof of a step of an incremental computation
-pub struct StepSNARK {
-  comm_T: CompressedCommitment,
+pub struct StepSNARK<G: Group> {
+  comm_T: Commitment<G>,
 }
 
-impl StepSNARK {
+impl<G: Group> StepSNARK<G> {
   fn protocol_name() -> &'static [u8] {
     b"NovaStepSNARK"
   }
@@ -38,16 +30,17 @@ /// and outputs a folded instance-witness tuple `(U, W)` of the same shape `shape`,
   /// with the guarantee that the folded witness `W` satisfies the folded instance `U`
   /// if and only if `W1` satisfies `U1` and `W2` satisfies `U2`.
   pub fn prove(
-    gens: &R1CSGens,
-    S: &R1CSShape,
-    U1: &R1CSInstance,
-    W1: &R1CSWitness,
-    U2: &R1CSInstance,
-    W2: &R1CSWitness,
+    gens: &R1CSGens<G>,
+    S: &R1CSShape<G>,
+    U1: &R1CSInstance<G>,
+    W1: &R1CSWitness<G>,
+    U2: &R1CSInstance<G>,
+    W2: &R1CSWitness<G>,
     transcript: &mut Transcript,
-  ) -> Result<(StepSNARK, (R1CSInstance, R1CSWitness)), NovaError> {
+  ) -> Result<(StepSNARK<G>, (R1CSInstance<G>, R1CSWitness<G>)), NovaError> {
     // append the protocol name to the transcript
-    transcript.append_protocol_name(StepSNARK::protocol_name());
+    //transcript.append_protocol_name(StepSNARK::protocol_name());
+    transcript.append_message(b"protocol-name", StepSNARK::<G>::protocol_name());
 
     // compute a commitment to the cross-term
     let (T, comm_T) = S.commit_T(gens, U1, W1, U2, W2)?;
@@ -56,10 +49,10 @@ // append `comm_T` to the transcript and obtain a challenge
     comm_T.append_to_transcript(b"comm_T", transcript);
 
     // compute a challenge from the transcript
-    let r = transcript.challenge_scalar(b"r");
+    let r = G::Scalar::challenge(b"r", transcript);
 
     // fold the instance using `r` and `comm_T`
-    let U = U1.fold(U2, &comm_T, &r)?;
+    let U = U1.fold(U2, &comm_T, &r);
 
     // fold the witness using `r` and `T`
     let W = W1.fold(W2, &T, &r)?;
@@ -75,21 +68,21 @@ /// with the guarantee that the folded instance `U`
   /// if and only if `U1` and `U2` are satisfiable.
   pub fn verify(
     &self,
-    U1: &R1CSInstance,
-    U2: &R1CSInstance,
+    U1: &R1CSInstance<G>,
+    U2: &R1CSInstance<G>,
     transcript: &mut Transcript,
-  ) -> Result<R1CSInstance, NovaError> {
+  ) -> Result<R1CSInstance<G>, NovaError> {
     // append the protocol name to the transcript
-    transcript.append_protocol_name(StepSNARK::protocol_name());
+    transcript.append_message(b"protocol-name", StepSNARK::<G>::protocol_name());
 
     // append `comm_T` to the transcript and obtain a challenge
     self.comm_T.append_to_transcript(b"comm_T", transcript);
 
     // compute a challenge from the transcript
-    let r = transcript.challenge_scalar(b"r");
+    let r = G::Scalar::challenge(b"r", transcript);
 
     // fold the instance using `r` and `comm_T`
-    let U = U1.fold(U2, &self.comm_T, &r)?;
+    let U = U1.fold(U2, &self.comm_T, &r);
 
     // return the folded instance
     Ok(U)
@@ -97,18 +90,23 @@ }
 }
 
 /// A SNARK that holds the proof of the final step of an incremental computation
-pub struct FinalSNARK {
-  W: R1CSWitness,
+pub struct FinalSNARK<G: Group> {
+  W: R1CSWitness<G>,
 }
 
-impl FinalSNARK {
+impl<G: Group> FinalSNARK<G> {
   /// Produces a proof of a instance given its satisfying witness `W`.
-  pub fn prove(W: &R1CSWitness) -> Result<FinalSNARK, NovaError> {
+  pub fn prove(W: &R1CSWitness<G>) -> Result<FinalSNARK<G>, NovaError> {
     Ok(FinalSNARK { W: W.clone() })
   }
 
   /// Verifies the proof of a folded instance `U` given its shape `S` public parameters `gens`
-  pub fn verify(&self, gens: &R1CSGens, S: &R1CSShape, U: &R1CSInstance) -> Result<(), NovaError> {
+  pub fn verify(
+    &self,
+    gens: &R1CSGens<G>,
+    S: &R1CSShape<G>,
+    U: &R1CSInstance<G>,
+  ) -> Result<(), NovaError> {
     // check that the witness is a valid witness to the folded instance `U`
     S.is_sat(gens, U, &self.W)
   }
@@ -116,13 +114,93 @@ }
 
 #[cfg(test)]
 mod tests {
-  use super::commitments::Scalar;
   use super::*;
-  use rand::rngs::OsRng;
+  use crate::traits::{CompressedGroup, Group, PrimeField};
+  use core::borrow::Borrow;
+  use curve25519_dalek::traits::MultiscalarMul;
+  use rand::{rngs::OsRng, CryptoRng, RngCore};
+
+  type S = curve25519_dalek::scalar::Scalar;
+  type G = curve25519_dalek::ristretto::RistrettoPoint;
+  type C = curve25519_dalek::ristretto::CompressedRistretto;
+
+  impl Group for G {
+    type Scalar = S;
+    type CompressedGroupElement = C;
+
+    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,
+    {
+      Self::multiscalar_mul(scalars, points)
+    }
+
+    fn compress(&self) -> Self::CompressedGroupElement {
+      self.compress()
+    }
+
+    fn from_uniform_bytes(bytes: &[u8]) -> Option<Self> {
+      if bytes.len() != 64 {
+        None
+      } else {
+        let mut arr = [0; 64];
+        arr.copy_from_slice(&bytes[0..64]);
+        Some(Self::from_uniform_bytes(&arr))
+      }
+    }
+  }
+
+  impl PrimeField for S {
+    fn zero() -> Self {
+      S::zero()
+    }
+
+    fn one() -> Self {
+      S::one()
+    }
+
+    fn from_bytes_mod_order_wide(bytes: &[u8]) -> Option<Self> {
+      if bytes.len() != 64 {
+        None
+      } else {
+        let mut arr = [0; 64];
+        arr.copy_from_slice(&bytes[0..64]);
+        Some(Self::from_bytes_mod_order_wide(&arr))
+      }
+    }
+
+    fn random(rng: &mut (impl RngCore + CryptoRng)) -> Self {
+      S::random(rng)
+    }
+  }
+
+  impl CompressedGroup for C {
+    type GroupElement = G;
+
+    fn decompress(&self) -> Option<Self::GroupElement> {
+      self.decompress()
+    }
+
+    fn as_bytes(&self) -> &[u8] {
+      self.as_bytes()
+    }
+  }
+
+  impl ChallengeTrait for S {
+    fn challenge(label: &'static [u8], transcript: &mut Transcript) -> Self {
+      let mut buf = [0u8; 64];
+      transcript.challenge_bytes(label, &mut buf);
+      S::from_bytes_mod_order_wide(&buf)
+    }
+  }
 
   #[test]
   fn test_tiny_r1cs() {
-    let one = Scalar::one();
+    let one = S::one();
     let (num_cons, num_vars, num_inputs, A, B, C) = {
       let num_cons = 4;
       let num_vars = 4;
@@ -138,9 +216,9 @@ // Relaxed R1CS is a set of three sparse matrices (A B C), where there is a row for every
       // constraint and a column for every entry in z = (vars, u, inputs)
       // An R1CS instance is satisfiable iff:
       // Az \circ Bz = u \cdot Cz + E, where z = (vars, 1, inputs)
-      let mut A: Vec<(usize, usize, Scalar)> = Vec::new();
-      let mut B: Vec<(usize, usize, Scalar)> = Vec::new();
-      let mut C: Vec<(usize, usize, Scalar)> = Vec::new();
+      let mut A: Vec<(usize, usize, S)> = Vec::new();
+      let mut B: Vec<(usize, usize, S)> = Vec::new();
+      let mut C: Vec<(usize, usize, S)> = Vec::new();
 
       // constraint 0 entries in (A,B,C)
       A.push((0, 0, one));
@@ -177,11 +255,11 @@
     // generate generators
     let gens = R1CSGens::new(num_cons, num_vars);
 
-    let rand_inst_witness_generator = |gens: &R1CSGens| -> (R1CSInstance, R1CSWitness) {
+    let rand_inst_witness_generator = |gens: &R1CSGens<G>| -> (R1CSInstance<G>, R1CSWitness<G>) {
       // compute a satisfying (vars, X) tuple
       let (vars, X) = {
         let mut csprng: OsRng = OsRng;
-        let z0 = Scalar::random(&mut csprng);
+        let z0 = S::random(&mut csprng);
         let z1 = z0 * z0; // constraint 0
         let z2 = z1 * z0; // constraint 1
         let z3 = z2 + z0; // constraint 2
@@ -193,14 +271,14 @@ (vars, X)
       };
 
       let W = {
-        let E = vec![Scalar::zero(); num_cons]; // default E
+        let E = vec![S::zero(); num_cons]; // default E
         let res = R1CSWitness::new(&S, &vars, &E);
         assert!(res.is_ok());
         res.unwrap()
       };
       let U = {
         let (comm_W, comm_E) = W.commit(&gens);
-        let u = Scalar::one(); //default u
+        let u = S::one(); //default u
         let res = R1CSInstance::new(&S, &comm_W, &comm_E, &X, &u);
         assert!(res.is_ok());
         res.unwrap()

@@ -1,41 +1,41 @@
 #![allow(clippy::type_complexity)]
-use super::commitments::Scalar;
-use super::commitments::{CommitGens, CommitTrait, Commitment, CompressedCommitment};
+use super::commitments::{CommitGens, CommitTrait, Commitment};
 use super::errors::NovaError;
+use super::traits::{Group, PrimeField};
 use itertools::concat;
 use rayon::prelude::*;
 
-pub struct R1CSGens {
-  gens_W: CommitGens,
-  gens_E: CommitGens,
+pub struct R1CSGens<G: Group> {
+  gens_W: CommitGens<G>,
+  gens_E: CommitGens<G>,
 }
 
 #[derive(Debug)]
-pub struct R1CSShape {
+pub struct R1CSShape<G: Group> {
   num_cons: usize,
   num_vars: usize,
   num_inputs: usize,
-  A: Vec<(usize, usize, Scalar)>,
-  B: Vec<(usize, usize, Scalar)>,
-  C: Vec<(usize, usize, Scalar)>,
+  A: Vec<(usize, usize, G::Scalar)>,
+  B: Vec<(usize, usize, G::Scalar)>,
+  C: Vec<(usize, usize, G::Scalar)>,
 }
 
 #[derive(Clone, Debug)]
-pub struct R1CSWitness {
-  W: Vec<Scalar>,
-  E: Vec<Scalar>,
+pub struct R1CSWitness<G: Group> {
+  W: Vec<G::Scalar>,
+  E: Vec<G::Scalar>,
 }
 
 #[derive(Clone, Debug, PartialEq, Eq)]
-pub struct R1CSInstance {
-  comm_W: Commitment,
-  comm_E: Commitment,
-  X: Vec<Scalar>,
-  u: Scalar,
+pub struct R1CSInstance<G: Group> {
+  comm_W: Commitment<G>,
+  comm_E: Commitment<G>,
+  X: Vec<G::Scalar>,
+  u: G::Scalar,
 }
 
-impl R1CSGens {
-  pub fn new(num_cons: usize, num_vars: usize) -> R1CSGens {
+impl<G: Group> R1CSGens<G> {
+  pub fn new(num_cons: usize, num_vars: usize) -> R1CSGens<G> {
     // generators to commit to witness vector `W`
     let gens_W = CommitGens::new(b"gens_W", num_vars);
 
@@ -46,19 +46,19 @@ R1CSGens { gens_E, gens_W }
   }
 }
 
-impl R1CSShape {
+impl<G: Group> R1CSShape<G> {
   pub fn new(
     num_cons: usize,
     num_vars: usize,
     num_inputs: usize,
-    A: &[(usize, usize, Scalar)],
-    B: &[(usize, usize, Scalar)],
-    C: &[(usize, usize, Scalar)],
-  ) -> Result<R1CSShape, NovaError> {
+    A: &[(usize, usize, G::Scalar)],
+    B: &[(usize, usize, G::Scalar)],
+    C: &[(usize, usize, G::Scalar)],
+  ) -> Result<R1CSShape<G>, NovaError> {
     let is_valid = |num_cons: usize,
                     num_vars: usize,
                     num_io: usize,
-                    M: &[(usize, usize, Scalar)]|
+                    M: &[(usize, usize, G::Scalar)]|
      -> Result<(), NovaError> {
       let res = (0..num_cons)
         .map(|i| {
@@ -100,8 +100,8 @@ }
 
   fn multiply_vec(
     &self,
-    z: &[Scalar],
-  ) -> Result<(Vec<Scalar>, Vec<Scalar>, Vec<Scalar>), NovaError> {
+    z: &[G::Scalar],
+  ) -> Result<(Vec<G::Scalar>, Vec<G::Scalar>, Vec<G::Scalar>), NovaError> {
     if z.len() != self.num_inputs + self.num_vars + 1 {
       return Err(NovaError::InvalidWitnessLength);
     }
@@ -110,13 +110,13 @@ // computes a product between a sparse matrix `M` and a vector `z`
     // This does not perform any validation of entries in M (e.g., if entries in `M` reference indexes outside the range of `z`)
     // This is safe since we know that `M` is valid
     let sparse_matrix_vec_product =
-      |M: &Vec<(usize, usize, Scalar)>, num_rows: usize, z: &[Scalar]| -> Vec<Scalar> {
+      |M: &Vec<(usize, usize, G::Scalar)>, num_rows: usize, z: &[G::Scalar]| -> Vec<G::Scalar> {
         (0..M.len())
           .map(|i| {
             let (row, col, val) = M[i];
             (row, val * z[col])
           })
-          .fold(vec![Scalar::zero(); num_rows], |mut Mz, (r, v)| {
+          .fold(vec![G::Scalar::zero(); num_rows], |mut Mz, (r, v)| {
             Mz[r] += v;
             Mz
           })
@@ -131,9 +131,9 @@ }
 
   pub fn is_sat(
     &self,
-    gens: &R1CSGens,
-    U: &R1CSInstance,
-    W: &R1CSWitness,
+    gens: &R1CSGens<G>,
+    U: &R1CSInstance<G>,
+    W: &R1CSWitness<G>,
   ) -> Result<(), NovaError> {
     assert_eq!(W.W.len(), self.num_vars);
     assert_eq!(W.E.len(), self.num_cons);
@@ -177,12 +177,12 @@ }
 
   pub fn commit_T(
     &self,
-    gens: &R1CSGens,
-    U1: &R1CSInstance,
-    W1: &R1CSWitness,
-    U2: &R1CSInstance,
-    W2: &R1CSWitness,
-  ) -> Result<(Vec<Scalar>, CompressedCommitment), NovaError> {
+    gens: &R1CSGens<G>,
+    U1: &R1CSInstance<G>,
+    W1: &R1CSWitness<G>,
+    U2: &R1CSInstance<G>,
+    W2: &R1CSWitness<G>,
+  ) -> Result<(Vec<G::Scalar>, Commitment<G>), NovaError> {
     let (AZ_1, BZ_1, CZ_1) = {
       let Z1 = concat(vec![W1.W.clone(), vec![U1.u], U1.X.clone()]);
       self.multiply_vec(&Z1)?
@@ -195,33 +195,37 @@ };
 
     let AZ_1_circ_BZ_2 = (0..AZ_1.len())
       .map(|i| AZ_1[i] * BZ_2[i])
-      .collect::<Vec<Scalar>>();
+      .collect::<Vec<G::Scalar>>();
     let AZ_2_circ_BZ_1 = (0..AZ_2.len())
       .map(|i| AZ_2[i] * BZ_1[i])
-      .collect::<Vec<Scalar>>();
+      .collect::<Vec<G::Scalar>>();
     let u_1_cdot_CZ_2 = (0..CZ_2.len())
       .map(|i| U1.u * CZ_2[i])
-      .collect::<Vec<Scalar>>();
+      .collect::<Vec<G::Scalar>>();
     let u_2_cdot_CZ_1 = (0..CZ_1.len())
       .map(|i| U2.u * CZ_1[i])
-      .collect::<Vec<Scalar>>();
+      .collect::<Vec<G::Scalar>>();
 
     let T = AZ_1_circ_BZ_2
       .par_iter()
       .zip(&AZ_2_circ_BZ_1)
       .zip(&u_1_cdot_CZ_2)
       .zip(&u_2_cdot_CZ_1)
-      .map(|(((a, b), c), d)| a + b - c - d)
-      .collect::<Vec<Scalar>>();
+      .map(|(((a, b), c), d)| *a + *b - *c - *d)
+      .collect::<Vec<G::Scalar>>();
 
-    let comm_T = T.commit(&gens.gens_E).compress();
+    let comm_T = T.commit(&gens.gens_E);
 
     Ok((T, comm_T))
   }
 }
 
-impl R1CSWitness {
-  pub fn new(S: &R1CSShape, W: &[Scalar], E: &[Scalar]) -> Result<R1CSWitness, NovaError> {
+impl<G: Group> R1CSWitness<G> {
+  pub fn new(
+    S: &R1CSShape<G>,
+    W: &[G::Scalar],
+    E: &[G::Scalar],
+  ) -> Result<R1CSWitness<G>, NovaError> {
     if S.num_vars != W.len() || S.num_cons != E.len() {
       Err(NovaError::InvalidWitnessLength)
     } else {
@@ -232,11 +236,16 @@ })
     }
   }
 
-  pub fn commit(&self, gens: &R1CSGens) -> (Commitment, Commitment) {
+  pub fn commit(&self, gens: &R1CSGens<G>) -> (Commitment<G>, Commitment<G>) {
     (self.W.commit(&gens.gens_W), self.E.commit(&gens.gens_E))
   }
 
-  pub fn fold(&self, W2: &R1CSWitness, T: &[Scalar], r: &Scalar) -> Result<R1CSWitness, NovaError> {
+  pub fn fold(
+    &self,
+    W2: &R1CSWitness<G>,
+    T: &[G::Scalar],
+    r: &G::Scalar,
+  ) -> Result<R1CSWitness<G>, NovaError> {
     let (W1, E1) = (&self.W, &self.E);
     let (W2, E2) = (&W2.W, &W2.E);
 
@@ -247,26 +256,26 @@
     let W = W1
       .par_iter()
       .zip(W2)
-      .map(|(a, b)| a + r * b)
-      .collect::<Vec<Scalar>>();
+      .map(|(a, b)| *a + *r * *b)
+      .collect::<Vec<G::Scalar>>();
     let E = E1
       .par_iter()
       .zip(T)
       .zip(E2)
-      .map(|((a, b), c)| a + r * b + r * r * c)
-      .collect::<Vec<Scalar>>();
+      .map(|((a, b), c)| *a + *r * *b + *r * *r * *c)
+      .collect::<Vec<G::Scalar>>();
     Ok(R1CSWitness { W, E })
   }
 }
 
-impl R1CSInstance {
+impl<G: Group> R1CSInstance<G> {
   pub fn new(
-    S: &R1CSShape,
-    comm_W: &Commitment,
-    comm_E: &Commitment,
-    X: &[Scalar],
-    u: &Scalar,
-  ) -> Result<R1CSInstance, NovaError> {
+    S: &R1CSShape<G>,
+    comm_W: &Commitment<G>,
+    comm_E: &Commitment<G>,
+    X: &[G::Scalar],
+    u: &G::Scalar,
+  ) -> Result<R1CSInstance<G>, NovaError> {
     if S.num_inputs != X.len() {
       Err(NovaError::InvalidInputLength)
     } else {
@@ -281,11 +290,10 @@ }
 
   pub fn fold(
     &self,
-    U2: &R1CSInstance,
-    comm_T: &CompressedCommitment,
-    r: &Scalar,
-  ) -> Result<R1CSInstance, NovaError> {
-    let comm_T_unwrapped = comm_T.decompress()?;
+    U2: &R1CSInstance<G>,
+    comm_T: &Commitment<G>,
+    r: &G::Scalar,
+  ) -> R1CSInstance<G> {
     let (X1, u1, comm_W_1, comm_E_1) =
       (&self.X, &self.u, &self.comm_W.clone(), &self.comm_E.clone());
     let (X2, u2, comm_W_2, comm_E_2) = (&U2.X, &U2.u, &U2.comm_W, &U2.comm_E);
@@ -294,17 +302,17 @@ // weighted sum of X
     let X = X1
       .par_iter()
       .zip(X2)
-      .map(|(a, b)| a + r * b)
-      .collect::<Vec<Scalar>>();
-    let comm_W = comm_W_1 + r * comm_W_2;
-    let comm_E = comm_E_1 + r * comm_T_unwrapped + r * r * comm_E_2;
-    let u = u1 + r * u2;
+      .map(|(a, b)| *a + *r * *b)
+      .collect::<Vec<G::Scalar>>();
+    let comm_W = comm_W_1 + *comm_W_2 * *r;
+    let comm_E = *comm_E_1 + *comm_T * *r + *comm_E_2 * *r * *r;
+    let u = *u1 + *r * *u2;
 
-    Ok(R1CSInstance {
+    R1CSInstance {
       comm_W,
       comm_E,
       X,
       u,
-    })
+    }
   }
 }

@@ -0,0 +1,88 @@
+use core::borrow::Borrow;
+use core::fmt::Debug;
+use core::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
+use merlin::Transcript;
+use rand::{CryptoRng, RngCore};
+
+/// Represents an element of a prime field
+pub trait PrimeField:
+  Sized
+  + Eq
+  + Copy
+  + Clone
+  + Default
+  + Send
+  + Sync
+  + Debug
+  + Add<Output = Self>
+  + Sub<Output = Self>
+  + Mul<Output = Self>
+  + Neg<Output = Self>
+  + for<'a> Add<&'a Self, Output = Self>
+  + for<'a> Mul<&'a Self, Output = Self>
+  + for<'a> Sub<&'a Self, Output = Self>
+  + AddAssign
+  + MulAssign
+  + SubAssign
+  + for<'a> AddAssign<&'a Self>
+  + for<'a> MulAssign<&'a Self>
+  + for<'a> SubAssign<&'a Self>
+{
+  /// returns the additive identity of the field
+  fn zero() -> Self;
+
+  /// returns the multiplicative identity of the field
+  fn one() -> Self;
+
+  /// converts the supplied bytes into an element of the field
+  fn from_bytes_mod_order_wide(bytes: &[u8]) -> Option<Self>;
+
+  /// returns an uniformly random element from the finite field
+  fn random(rng: &mut (impl RngCore + CryptoRng)) -> Self;
+}
+
+/// Represents an element of a group
+pub trait Group:
+  Clone
+  + Copy
+  + Debug
+  + Eq
+  + Sized
+  + Mul<<Self as Group>::Scalar, Output = Self>
+  + MulAssign<<Self as Group>::Scalar>
+  + for<'a> MulAssign<&'a <Self as Group>::Scalar>
+  + for<'a> Mul<&'a <Self as Group>::Scalar, Output = Self>
+  + Add<Self, Output = Self>
+  + AddAssign<Self>
+  + for<'a> AddAssign<&'a Self>
+  + for<'a> Add<&'a Self, Output = Self>
+{
+  type Scalar: PrimeField + ChallengeTrait;
+  type CompressedGroupElement: CompressedGroup;
+
+  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;
+
+  fn compress(&self) -> Self::CompressedGroupElement;
+
+  fn from_uniform_bytes(bytes: &[u8]) -> Option<Self>;
+}
+
+/// Represents a compressed version of a group element
+pub trait CompressedGroup: Clone + Copy + Debug + Eq + Sized + Send + Sync + 'static {
+  type GroupElement: Group;
+
+  fn decompress(&self) -> Option<Self::GroupElement>;
+
+  fn as_bytes(&self) -> &[u8];
+}
+
+/// A helper trait to generate challenges using a transcript object
+pub trait ChallengeTrait {
+  fn challenge(label: &'static [u8], transcript: &mut Transcript) -> Self;
+}