diff --git a/src/commitments.rs b/src/commitments.rs
index 89eae87..8d64f2b 100644
--- a/src/commitments.rs
+++ b/src/commitments.rs
@@ -94,8 +94,18 @@ impl<'b, G: Group> MulAssign<&'b G::Scalar> for Commitment<G> {
   }
 }
 
+impl<'a, 'b, G: Group> Mul<&'b G::Scalar> for &'a Commitment<G> {
+  type Output = Commitment<G>;
+  fn mul(self, scalar: &'b G::Scalar) -> Commitment<G> {
+    Commitment {
+      comm: self.comm * scalar,
+    }
+  }
+}
+
 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,
@@ -119,29 +129,40 @@ impl<'a, 'b, G: Group> Add<&'b Commitment<G>> for &'a Commitment<G> {
   }
 }
 
-impl<G: Group> AddAssign<Commitment<G>> for Commitment<G> {
-  fn add_assign(&mut self, rhs: Commitment<G>) {
-    *self += &rhs;
-  }
-}
+macro_rules! define_add_variants {
+  (G = $g:path, LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {
+    impl<'b, G: $g> Add<&'b $rhs> for $lhs {
+      type Output = $out;
+      fn add(self, rhs: &'b $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
-  }
-}
+    impl<'a, G: $g> Add<$rhs> for &'a $lhs {
+      type Output = $out;
+      fn add(self, rhs: $rhs) -> $out {
+        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
-  }
+    impl<G: $g> Add<$rhs> for $lhs {
+      type Output = $out;
+      fn add(self, rhs: $rhs) -> $out {
+        &self + &rhs
+      }
+    }
+  };
 }
 
-impl<G: Group> Add<Commitment<G>> for Commitment<G> {
-  type Output = Commitment<G>;
-  fn add(self, rhs: Commitment<G>) -> Commitment<G> {
-    &self + &rhs
-  }
+macro_rules! define_add_assign_variants {
+  (G = $g:path, LHS = $lhs:ty, RHS = $rhs:ty) => {
+    impl<G: $g> AddAssign<$rhs> for $lhs {
+      fn add_assign(&mut self, rhs: $rhs) {
+        *self += &rhs;
+      }
+    }
+  };
 }
+
+define_add_assign_variants!(G = Group, LHS = Commitment<G>, RHS = Commitment<G>);
+define_add_variants!(G = Group, LHS = Commitment<G>, RHS = Commitment<G>, Output = Commitment<G>);
diff --git a/src/lib.rs b/src/lib.rs
index 0b5a4e1..a0768ac 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -8,7 +8,9 @@ mod errors;
 mod r1cs;
 mod traits;
 
-use commitments::{AppendToTranscriptTrait, Commitment};
+use std::marker::PhantomData;
+
+use commitments::{AppendToTranscriptTrait, CompressedCommitment};
 use errors::NovaError;
 use merlin::Transcript;
 use r1cs::{R1CSGens, R1CSInstance, R1CSShape, R1CSWitness};
@@ -16,7 +18,8 @@ use traits::{ChallengeTrait, Group};
 
 /// A SNARK that holds the proof of a step of an incremental computation
 pub struct StepSNARK<G: Group> {
-  comm_T: Commitment<G>,
+  comm_T: CompressedCommitment<G::CompressedGroupElement>,
+  _p: PhantomData<G>,
 }
 
 impl<G: Group> StepSNARK<G> {
@@ -52,13 +55,19 @@ impl<G: Group> StepSNARK<G> {
     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)?;
 
     // return the folded instance and witness
-    Ok((StepSNARK { comm_T }, (U, W)))
+    Ok((
+      StepSNARK {
+        comm_T,
+        _p: Default::default(),
+      },
+      (U, W),
+    ))
   }
 
   /// Takes as input two relaxed R1CS instances `U1` and `U2`
@@ -82,7 +91,7 @@ impl<G: Group> StepSNARK<G> {
     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)
diff --git a/src/r1cs.rs b/src/r1cs.rs
index 980b6df..bd95484 100644
--- a/src/r1cs.rs
+++ b/src/r1cs.rs
@@ -1,4 +1,6 @@
 #![allow(clippy::type_complexity)]
+use crate::commitments::CompressedCommitment;
+
 use super::commitments::{CommitGens, CommitTrait, Commitment};
 use super::errors::NovaError;
 use super::traits::{Group, PrimeField};
@@ -182,7 +184,13 @@ impl<G: Group> R1CSShape<G> {
     W1: &R1CSWitness<G>,
     U2: &R1CSInstance<G>,
     W2: &R1CSWitness<G>,
-  ) -> Result<(Vec<G::Scalar>, Commitment<G>), NovaError> {
+  ) -> Result<
+    (
+      Vec<G::Scalar>,
+      CompressedCommitment<G::CompressedGroupElement>,
+    ),
+    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)?
@@ -214,7 +222,7 @@ impl<G: Group> R1CSShape<G> {
       .map(|(((a, b), c), d)| *a + *b - *c - *d)
       .collect::<Vec<G::Scalar>>();
 
-    let comm_T = T.commit(&gens.gens_E);
+    let comm_T = T.commit(&gens.gens_E).compress();
 
     Ok((T, comm_T))
   }
@@ -291,9 +299,10 @@ impl<G: Group> R1CSInstance<G> {
   pub fn fold(
     &self,
     U2: &R1CSInstance<G>,
-    comm_T: &Commitment<G>,
+    comm_T: &CompressedCommitment<G::CompressedGroupElement>,
     r: &G::Scalar,
-  ) -> R1CSInstance<G> {
+  ) -> Result<R1CSInstance<G>, NovaError> {
+    let comm_T_unwrapped = comm_T.decompress()?;
     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);
@@ -304,15 +313,15 @@ impl<G: Group> R1CSInstance<G> {
       .zip(X2)
       .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 comm_W = comm_W_1 + comm_W_2 * r;
+    let comm_E = *comm_E_1 + comm_T_unwrapped * *r + comm_E_2 * r * *r;
     let u = *u1 + *r * *u2;
 
-    R1CSInstance {
+    Ok(R1CSInstance {
       comm_W,
       comm_E,
       X,
       u,
-    }
+    })
   }
 }
diff --git a/src/traits.rs b/src/traits.rs
index 57ddc7e..15d4274 100644
--- a/src/traits.rs
+++ b/src/traits.rs
@@ -48,17 +48,13 @@ pub trait Group:
   + 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>
+  + GroupOps
+  + GroupOpsOwned
+  + ScalarMul<<Self as Group>::Scalar>
+  + ScalarMulOwned<<Self as Group>::Scalar>
 {
   type Scalar: PrimeField + ChallengeTrait;
-  type CompressedGroupElement: CompressedGroup;
+  type CompressedGroupElement: CompressedGroup<GroupElement = Self>;
 
   fn vartime_multiscalar_mul<I, J>(scalars: I, points: J) -> Self
   where
@@ -86,3 +82,28 @@ pub trait CompressedGroup: Clone + Copy + Debug + Eq + Sized + Send + Sync + 'st
 pub trait ChallengeTrait {
   fn challenge(label: &'static [u8], transcript: &mut Transcript) -> Self;
 }
+
+/// A helper trait for types with a group operation.
+pub trait GroupOps<Rhs = Self, Output = Self>:
+  Add<Rhs, Output = Output> + Sub<Rhs, Output = Output> + AddAssign<Rhs> + SubAssign<Rhs>
+{
+}
+
+impl<T, Rhs, Output> GroupOps<Rhs, Output> for T where
+  T: Add<Rhs, Output = Output> + Sub<Rhs, Output = Output> + AddAssign<Rhs> + SubAssign<Rhs>
+{
+}
+
+/// A helper trait for references with a group operation.
+pub trait GroupOpsOwned<Rhs = Self, Output = Self>: for<'r> GroupOps<&'r Rhs, Output> {}
+impl<T, Rhs, Output> GroupOpsOwned<Rhs, Output> for T where T: for<'r> GroupOps<&'r Rhs, Output> {}
+
+/// A helper trait for types implementing group scalar multiplication.
+pub trait ScalarMul<Rhs, Output = Self>: Mul<Rhs, Output = Output> + MulAssign<Rhs> {}
+
+impl<T, Rhs, Output> ScalarMul<Rhs, Output> for T where T: Mul<Rhs, Output = Output> + MulAssign<Rhs>
+{}
+
+/// A helper trait for references implementing group scalar multiplication.
+pub trait ScalarMulOwned<Rhs, Output = Self>: for<'r> ScalarMul<&'r Rhs, Output> {}
+impl<T, Rhs, Output> ScalarMulOwned<Rhs, Output> for T where T: for<'r> ScalarMul<&'r Rhs, Output> {}