diff --git a/bellman/src/domain.rs b/bellman/src/domain.rs
index 47f8ff083..e1dc22970 100644
--- a/bellman/src/domain.rs
+++ b/bellman/src/domain.rs
@@ -12,7 +12,7 @@
 //! [Groth16]: https://eprint.iacr.org/2016/260
 
 use ff::PrimeField;
-use group::CurveProjective;
+use group::CofactorCurve;
 
 use super::SynthesisError;
 
@@ -196,23 +196,23 @@ pub trait Group<Scalar: PrimeField>: Sized + Copy + Clone + Send + Sync {
     fn group_sub_assign(&mut self, other: &Self);
 }
 
-pub struct Point<G: CurveProjective>(pub G);
+pub struct Point<G: CofactorCurve>(pub G);
 
-impl<G: CurveProjective> PartialEq for Point<G> {
+impl<G: CofactorCurve> PartialEq for Point<G> {
     fn eq(&self, other: &Point<G>) -> bool {
         self.0 == other.0
     }
 }
 
-impl<G: CurveProjective> Copy for Point<G> {}
+impl<G: CofactorCurve> Copy for Point<G> {}
 
-impl<G: CurveProjective> Clone for Point<G> {
+impl<G: CofactorCurve> Clone for Point<G> {
     fn clone(&self) -> Point<G> {
         *self
     }
 }
 
-impl<G: CurveProjective> Group<G::Scalar> for Point<G> {
+impl<G: CofactorCurve> Group<G::Scalar> for Point<G> {
     fn group_zero() -> Self {
         Point(G::identity())
     }
diff --git a/bellman/src/groth16/generator.rs b/bellman/src/groth16/generator.rs
index 060ff1bec..e84fc12a9 100644
--- a/bellman/src/groth16/generator.rs
+++ b/bellman/src/groth16/generator.rs
@@ -3,7 +3,7 @@ use std::ops::{AddAssign, MulAssign};
 use std::sync::Arc;
 
 use ff::{Field, PrimeField};
-use group::{CurveAffine, CurveProjective, Group, Wnaf};
+use group::{CurveAffine, CofactorCurve, Group, Wnaf};
 use pairing::Engine;
 
 use super::{Parameters, VerifyingKey};
diff --git a/bellman/src/groth16/prover.rs b/bellman/src/groth16/prover.rs
index ff91c2675..293cad2b7 100644
--- a/bellman/src/groth16/prover.rs
+++ b/bellman/src/groth16/prover.rs
@@ -5,7 +5,7 @@ use std::sync::Arc;
 use futures::Future;
 
 use ff::{Field, PrimeField};
-use group::{CurveAffine, CurveProjective};
+use group::{CurveAffine, CofactorCurve};
 use pairing::Engine;
 
 use super::{ParameterSource, Proof};
diff --git a/bellman/src/groth16/tests/dummy_engine.rs b/bellman/src/groth16/tests/dummy_engine.rs
index 3ddb98c90..01e8b82d1 100644
--- a/bellman/src/groth16/tests/dummy_engine.rs
+++ b/bellman/src/groth16/tests/dummy_engine.rs
@@ -1,5 +1,5 @@
 use ff::{Field, PrimeField};
-use group::{CurveAffine, CurveProjective, Group, GroupEncoding, PrimeGroup, UncompressedEncoding};
+use group::{CurveAffine, CofactorCurve, Group, GroupEncoding, PrimeGroup, UncompressedEncoding};
 use pairing::{Engine, MillerLoopResult, MultiMillerLoop, PairingCurveAffine};
 
 use rand_core::RngCore;
@@ -393,7 +393,7 @@ impl Group for Fr {
 
 impl PrimeGroup for Fr {}
 
-impl CurveProjective for Fr {
+impl CofactorCurve for Fr {
     type Affine = Fr;
 
     fn to_affine(&self) -> Fr {
@@ -425,7 +425,7 @@ impl AsRef<[u8]> for FakePoint {
 }
 
 impl CurveAffine for Fr {
-    type Projective = Fr;
+    type Curve = Fr;
     type Scalar = Fr;
 
     fn identity() -> Self {
@@ -440,7 +440,7 @@ impl CurveAffine for Fr {
         Choice::from(if <Fr as Field>::is_zero(self) { 1 } else { 0 })
     }
 
-    fn to_projective(&self) -> Self::Projective {
+    fn to_curve(&self) -> Self::Curve {
         *self
     }
 }
diff --git a/bellman/src/groth16/verifier.rs b/bellman/src/groth16/verifier.rs
index e86b15aa5..5758d5d83 100644
--- a/bellman/src/groth16/verifier.rs
+++ b/bellman/src/groth16/verifier.rs
@@ -1,4 +1,4 @@
-use group::{CurveAffine, CurveProjective};
+use group::{CurveAffine, CofactorCurve};
 use pairing::{MillerLoopResult, MultiMillerLoop};
 use std::ops::{AddAssign, Neg};
 
@@ -27,7 +27,7 @@ pub fn verify_proof<'a, E: MultiMillerLoop>(
         return Err(SynthesisError::MalformedVerifyingKey);
     }
 
-    let mut acc = pvk.ic[0].to_projective();
+    let mut acc = pvk.ic[0].to_curve();
 
     for (i, b) in public_inputs.iter().zip(pvk.ic.iter().skip(1)) {
         AddAssign::<&E::G1>::add_assign(&mut acc, &(*b * i));
diff --git a/bellman/src/multiexp.rs b/bellman/src/multiexp.rs
index c9e00d6ba..fea10c667 100644
--- a/bellman/src/multiexp.rs
+++ b/bellman/src/multiexp.rs
@@ -2,7 +2,7 @@ use super::multicore::Worker;
 use bit_vec::{self, BitVec};
 use ff::{Endianness, Field, PrimeField};
 use futures::Future;
-use group::{CurveAffine, CurveProjective};
+use group::{CofactorCurve, CurveAffine};
 use std::io;
 use std::iter;
 use std::ops::AddAssign;
@@ -25,17 +25,17 @@ pub trait Source<G: CurveAffine> {
     fn skip(&mut self, amt: usize) -> Result<(), SynthesisError>;
 }
 
-pub trait AddAssignFromSource: CurveProjective {
+pub trait AddAssignFromSource: CofactorCurve {
     /// Parses the element from the source. Fails if the point is at infinity.
-    fn add_assign_from_source<S: Source<<Self as CurveProjective>::Affine>>(
+    fn add_assign_from_source<S: Source<<Self as CofactorCurve>::Affine>>(
         &mut self,
         source: &mut S,
     ) -> Result<(), SynthesisError> {
-        AddAssign::<&<Self as CurveProjective>::Affine>::add_assign(self, source.next()?);
+        AddAssign::<&<Self as CofactorCurve>::Affine>::add_assign(self, source.next()?);
         Ok(())
     }
 }
-impl<G> AddAssignFromSource for G where G: CurveProjective {}
+impl<G> AddAssignFromSource for G where G: CofactorCurve {}
 
 impl<G: CurveAffine> SourceBuilder<G> for (Arc<Vec<G>>, usize) {
     type Source = (Arc<Vec<G>>, usize);
@@ -162,8 +162,8 @@ fn multiexp_inner<Q, D, G, S>(
 where
     for<'a> &'a Q: QueryDensity,
     D: Send + Sync + 'static + Clone + AsRef<Q>,
-    G: CurveProjective,
-    S: SourceBuilder<<G as CurveProjective>::Affine>,
+    G: CofactorCurve,
+    S: SourceBuilder<<G as CofactorCurve>::Affine>,
 {
     // Perform this region of the multiexp
     let this = {
@@ -274,8 +274,8 @@ pub fn multiexp<Q, D, G, S>(
 where
     for<'a> &'a Q: QueryDensity,
     D: Send + Sync + 'static + Clone + AsRef<Q>,
-    G: CurveProjective,
-    S: SourceBuilder<<G as CurveProjective>::Affine>,
+    G: CofactorCurve,
+    S: SourceBuilder<<G as CofactorCurve>::Affine>,
 {
     let c = if exponents.len() < 32 {
         3u32
@@ -296,8 +296,8 @@ where
 #[cfg(feature = "pairing")]
 #[test]
 fn test_with_bls12() {
-    fn naive_multiexp<G: CurveProjective>(
-        bases: Arc<Vec<<G as CurveProjective>::Affine>>,
+    fn naive_multiexp<G: CofactorCurve>(
+        bases: Arc<Vec<<G as CofactorCurve>::Affine>>,
         exponents: Arc<Vec<G::Scalar>>,
     ) -> G {
         assert_eq!(bases.len(), exponents.len());
diff --git a/group/src/lib.rs b/group/src/lib.rs
index 0b105c051..839e31a3b 100644
--- a/group/src/lib.rs
+++ b/group/src/lib.rs
@@ -88,12 +88,10 @@ pub trait PrimeGroup: Group {}
 
 /// Projective representation of an elliptic curve point guaranteed to be
 /// in the correct prime order subgroup.
-pub trait CurveProjective:
-    Group
-    + GroupOps<<Self as CurveProjective>::Affine>
-    + GroupOpsOwned<<Self as CurveProjective>::Affine>
+pub trait CofactorCurve:
+    Group + GroupOps<<Self as CofactorCurve>::Affine> + GroupOpsOwned<<Self as CofactorCurve>::Affine>
 {
-    type Affine: CurveAffine<Projective = Self, Scalar = Self::Scalar>
+    type Affine: CurveAffine<Curve = Self, Scalar = Self::Scalar>
         + Mul<Self::Scalar, Output = Self>
         + for<'r> Mul<Self::Scalar, Output = Self>;
 
@@ -134,11 +132,11 @@ pub trait CurveAffine:
     + 'static
     + GroupEncoding
     + Neg<Output = Self>
-    + Mul<<Self as CurveAffine>::Scalar, Output = <Self as CurveAffine>::Projective>
-    + for<'r> Mul<<Self as CurveAffine>::Scalar, Output = <Self as CurveAffine>::Projective>
+    + Mul<<Self as CurveAffine>::Scalar, Output = <Self as CurveAffine>::Curve>
+    + for<'r> Mul<<Self as CurveAffine>::Scalar, Output = <Self as CurveAffine>::Curve>
 {
     type Scalar: PrimeField;
-    type Projective: CurveProjective<Affine = Self, Scalar = Self::Scalar>;
+    type Curve: CofactorCurve<Affine = Self, Scalar = Self::Scalar>;
 
     /// Returns the additive identity.
     fn identity() -> Self;
@@ -150,8 +148,8 @@ pub trait CurveAffine:
     /// additive identity.
     fn is_identity(&self) -> Choice;
 
-    /// Converts this element into its affine representation.
-    fn to_projective(&self) -> Self::Projective;
+    /// Converts this element into its efficient representation.
+    fn to_curve(&self) -> Self::Curve;
 }
 
 pub trait GroupEncoding: Sized {
diff --git a/group/src/tests/mod.rs b/group/src/tests/mod.rs
index 763cee28e..920b44c91 100644
--- a/group/src/tests/mod.rs
+++ b/group/src/tests/mod.rs
@@ -3,9 +3,9 @@ use rand::SeedableRng;
 use rand_xorshift::XorShiftRng;
 use std::ops::{Mul, Neg};
 
-use crate::{CurveAffine, CurveProjective, GroupEncoding, UncompressedEncoding};
+use crate::{CofactorCurve, CurveAffine, GroupEncoding, UncompressedEncoding};
 
-pub fn curve_tests<G: CurveProjective>() {
+pub fn curve_tests<G: CofactorCurve>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -50,8 +50,8 @@ pub fn curve_tests<G: CurveProjective>() {
     // Transformations
     {
         let a = G::random(&mut rng);
-        let b = a.to_affine().to_projective();
-        let c = a.to_affine().to_projective().to_affine().to_projective();
+        let b = a.to_affine().to_curve();
+        let c = a.to_affine().to_curve().to_affine().to_curve();
         assert_eq!(a, b);
         assert_eq!(b, c);
     }
@@ -65,7 +65,7 @@ pub fn curve_tests<G: CurveProjective>() {
     random_compressed_encoding_tests::<G>();
 }
 
-fn random_wnaf_tests<G: CurveProjective>() {
+fn random_wnaf_tests<G: CofactorCurve>() {
     use crate::wnaf::*;
 
     let mut rng = XorShiftRng::from_seed([
@@ -184,7 +184,7 @@ fn random_wnaf_tests<G: CurveProjective>() {
     }
 }
 
-fn random_negation_tests<G: CurveProjective>() {
+fn random_negation_tests<G: CofactorCurve>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -214,7 +214,7 @@ fn random_negation_tests<G: CurveProjective>() {
     }
 }
 
-fn random_doubling_tests<G: CurveProjective>() {
+fn random_doubling_tests<G: CofactorCurve>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -242,7 +242,7 @@ fn random_doubling_tests<G: CurveProjective>() {
     }
 }
 
-fn random_multiplication_tests<G: CurveProjective>() {
+fn random_multiplication_tests<G: CofactorCurve>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -277,7 +277,7 @@ fn random_multiplication_tests<G: CurveProjective>() {
     }
 }
 
-fn random_addition_tests<G: CurveProjective>() {
+fn random_addition_tests<G: CofactorCurve>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -325,17 +325,17 @@ fn random_addition_tests<G: CurveProjective>() {
         // Mixed addition
 
         // (a + b) + c
-        tmp[3] = a_affine.to_projective();
+        tmp[3] = a_affine.to_curve();
         tmp[3].add_assign(&b_affine);
         tmp[3].add_assign(&c_affine);
 
         // a + (b + c)
-        tmp[4] = b_affine.to_projective();
+        tmp[4] = b_affine.to_curve();
         tmp[4].add_assign(&c_affine);
         tmp[4].add_assign(&a_affine);
 
         // (a + c) + b
-        tmp[5] = a_affine.to_projective();
+        tmp[5] = a_affine.to_curve();
         tmp[5].add_assign(&c_affine);
         tmp[5].add_assign(&b_affine);
 
@@ -357,7 +357,7 @@ fn random_addition_tests<G: CurveProjective>() {
     }
 }
 
-fn random_transformation_tests<G: CurveProjective>() {
+fn random_transformation_tests<G: CofactorCurve>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -366,7 +366,7 @@ fn random_transformation_tests<G: CurveProjective>() {
     for _ in 0..1000 {
         let g = G::random(&mut rng);
         let g_affine = g.to_affine();
-        let g_projective = g_affine.to_projective();
+        let g_projective = g_affine.to_curve();
         assert_eq!(g, g_projective);
     }
 
@@ -382,7 +382,7 @@ fn random_transformation_tests<G: CurveProjective>() {
         }
         for _ in 0..5 {
             let s = between.sample(&mut rng);
-            v[s] = v[s].to_affine().to_projective();
+            v[s] = v[s].to_affine().to_curve();
         }
 
         let expected_v = v.iter().map(|v| v.to_affine()).collect::<Vec<_>>();
@@ -394,7 +394,7 @@ fn random_transformation_tests<G: CurveProjective>() {
     }
 }
 
-fn random_compressed_encoding_tests<G: CurveProjective>() {
+fn random_compressed_encoding_tests<G: CofactorCurve>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -420,9 +420,9 @@ fn random_compressed_encoding_tests<G: CurveProjective>() {
     }
 }
 
-pub fn random_uncompressed_encoding_tests<G: CurveProjective>()
+pub fn random_uncompressed_encoding_tests<G: CofactorCurve>()
 where
-    G::Affine: UncompressedEncoding,
+    <G as CofactorCurve>::Affine: UncompressedEncoding,
 {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
diff --git a/group/src/wnaf.rs b/group/src/wnaf.rs
index 0ee46cafc..64a2de1b5 100644
--- a/group/src/wnaf.rs
+++ b/group/src/wnaf.rs
@@ -2,10 +2,10 @@ use byteorder::{ByteOrder, LittleEndian};
 use ff::PrimeField;
 use std::iter;
 
-use super::{CurveProjective, Group};
+use super::{CofactorCurve, Group};
 
 /// Replaces the contents of `table` with a w-NAF window table for the given window size.
-pub(crate) fn wnaf_table<G: CurveProjective>(table: &mut Vec<G>, mut base: G, window: usize) {
+pub(crate) fn wnaf_table<G: CofactorCurve>(table: &mut Vec<G>, mut base: G, window: usize) {
     table.truncate(0);
     table.reserve(1 << (window - 1));
 
@@ -78,7 +78,7 @@ pub(crate) fn wnaf_form<S: AsRef<[u8]>>(wnaf: &mut Vec<i64>, c: S, window: usize
 ///
 /// This function must be provided a `table` and `wnaf` that were constructed with
 /// the same window size; otherwise, it may panic or produce invalid results.
-pub(crate) fn wnaf_exp<G: CurveProjective>(table: &[G], wnaf: &[i64]) -> G {
+pub(crate) fn wnaf_exp<G: CofactorCurve>(table: &[G], wnaf: &[i64]) -> G {
     let mut result = G::identity();
 
     let mut found_one = false;
@@ -110,7 +110,7 @@ pub struct Wnaf<W, B, S> {
     window_size: W,
 }
 
-impl<G: CurveProjective> Wnaf<(), Vec<G>, Vec<i64>> {
+impl<G: CofactorCurve> Wnaf<(), Vec<G>, Vec<i64>> {
     /// Construct a new wNAF context without allocating.
     pub fn new() -> Self {
         Wnaf {
@@ -157,7 +157,7 @@ impl<G: CurveProjective> Wnaf<(), Vec<G>, Vec<i64>> {
     }
 }
 
-impl<'a, G: CurveProjective> Wnaf<usize, &'a [G], &'a mut Vec<i64>> {
+impl<'a, G: CofactorCurve> Wnaf<usize, &'a [G], &'a mut Vec<i64>> {
     /// Constructs new space for the scalar representation while borrowing
     /// the computed window table, for sending the window table across threads.
     pub fn shared(&self) -> Wnaf<usize, &'a [G], Vec<i64>> {
@@ -169,7 +169,7 @@ impl<'a, G: CurveProjective> Wnaf<usize, &'a [G], &'a mut Vec<i64>> {
     }
 }
 
-impl<'a, G: CurveProjective> Wnaf<usize, &'a mut Vec<G>, &'a [i64]> {
+impl<'a, G: CofactorCurve> Wnaf<usize, &'a mut Vec<G>, &'a [i64]> {
     /// Constructs new space for the window table while borrowing
     /// the computed scalar representation, for sending the scalar representation
     /// across threads.
@@ -184,7 +184,7 @@ impl<'a, G: CurveProjective> Wnaf<usize, &'a mut Vec<G>, &'a [i64]> {
 
 impl<B, S: AsRef<[i64]>> Wnaf<usize, B, S> {
     /// Performs exponentiation given a base.
-    pub fn base<G: CurveProjective>(&mut self, base: G) -> G
+    pub fn base<G: CofactorCurve>(&mut self, base: G) -> G
     where
         B: AsMut<Vec<G>>,
     {
@@ -195,7 +195,7 @@ impl<B, S: AsRef<[i64]>> Wnaf<usize, B, S> {
 
 impl<B, S: AsMut<Vec<i64>>> Wnaf<usize, B, S> {
     /// Performs exponentiation given a scalar.
-    pub fn scalar<G: CurveProjective>(&mut self, scalar: &<G as Group>::Scalar) -> G
+    pub fn scalar<G: CofactorCurve>(&mut self, scalar: &<G as Group>::Scalar) -> G
     where
         B: AsRef<[G]>,
     {
diff --git a/pairing/src/bls12_381/ec.rs b/pairing/src/bls12_381/ec.rs
index 8d2a3f18c..e480700f8 100644
--- a/pairing/src/bls12_381/ec.rs
+++ b/pairing/src/bls12_381/ec.rs
@@ -199,7 +199,7 @@ macro_rules! curve_impl {
 
         impl CurveAffine for $affine {
             type Scalar = $scalarfield;
-            type Projective = $projective;
+            type Curve = $projective;
 
             fn identity() -> Self {
                 $affine {
@@ -217,7 +217,7 @@ macro_rules! curve_impl {
                 Choice::from(if self.infinity { 1 } else { 0 })
             }
 
-            fn to_projective(&self) -> $projective {
+            fn to_curve(&self) -> $projective {
                 (*self).into()
             }
         }
@@ -466,30 +466,28 @@ macro_rules! curve_impl {
             }
         }
 
-        impl<'r> ::std::ops::Add<&'r <$projective as CurveProjective>::Affine> for $projective {
+        impl<'r> ::std::ops::Add<&'r $affine> for $projective {
             type Output = Self;
 
             #[inline]
-            fn add(self, other: &<$projective as CurveProjective>::Affine) -> Self {
+            fn add(self, other: &$affine) -> Self {
                 let mut ret = self;
                 ret.add_assign(other);
                 ret
             }
         }
 
-        impl ::std::ops::Add<<$projective as CurveProjective>::Affine> for $projective {
+        impl ::std::ops::Add<$affine> for $projective {
             type Output = Self;
 
             #[inline]
-            fn add(self, other: <$projective as CurveProjective>::Affine) -> Self {
+            fn add(self, other: $affine) -> Self {
                 self + &other
             }
         }
 
-        impl<'r> ::std::ops::AddAssign<&'r <$projective as CurveProjective>::Affine>
-            for $projective
-        {
-            fn add_assign(&mut self, other: &<$projective as CurveProjective>::Affine) {
+        impl<'r> ::std::ops::AddAssign<&'r $affine> for $projective {
+            fn add_assign(&mut self, other: &$affine) {
                 if other.is_identity().into() {
                     return;
                 }
@@ -567,44 +565,42 @@ macro_rules! curve_impl {
             }
         }
 
-        impl ::std::ops::AddAssign<<$projective as CurveProjective>::Affine> for $projective {
+        impl ::std::ops::AddAssign<$affine> for $projective {
             #[inline]
-            fn add_assign(&mut self, other: <$projective as CurveProjective>::Affine) {
+            fn add_assign(&mut self, other: $affine) {
                 self.add_assign(&other);
             }
         }
 
-        impl<'r> ::std::ops::Sub<&'r <$projective as CurveProjective>::Affine> for $projective {
+        impl<'r> ::std::ops::Sub<&'r $affine> for $projective {
             type Output = Self;
 
             #[inline]
-            fn sub(self, other: &<$projective as CurveProjective>::Affine) -> Self {
+            fn sub(self, other: &$affine) -> Self {
                 let mut ret = self;
                 ret.sub_assign(other);
                 ret
             }
         }
 
-        impl ::std::ops::Sub<<$projective as CurveProjective>::Affine> for $projective {
+        impl ::std::ops::Sub<$affine> for $projective {
             type Output = Self;
 
             #[inline]
-            fn sub(self, other: <$projective as CurveProjective>::Affine) -> Self {
+            fn sub(self, other: $affine) -> Self {
                 self - &other
             }
         }
 
-        impl<'r> ::std::ops::SubAssign<&'r <$projective as CurveProjective>::Affine>
-            for $projective
-        {
-            fn sub_assign(&mut self, other: &<$projective as CurveProjective>::Affine) {
+        impl<'r> ::std::ops::SubAssign<&'r $affine> for $projective {
+            fn sub_assign(&mut self, other: &$affine) {
                 self.add_assign(&other.neg());
             }
         }
 
-        impl ::std::ops::SubAssign<<$projective as CurveProjective>::Affine> for $projective {
+        impl ::std::ops::SubAssign<$affine> for $projective {
             #[inline]
-            fn sub_assign(&mut self, other: <$projective as CurveProjective>::Affine) {
+            fn sub_assign(&mut self, other: $affine) {
                 self.sub_assign(&other);
             }
         }
@@ -746,7 +742,7 @@ macro_rules! curve_impl {
 
         impl PrimeGroup for $projective {}
 
-        impl CurveProjective for $projective {
+        impl CofactorCurve for $projective {
             type Affine = $affine;
 
             fn batch_normalize(p: &[Self], q: &mut [$affine]) {
@@ -908,7 +904,7 @@ pub mod g1 {
     use crate::{Engine, PairingCurveAffine};
     use ff::{BitIterator, Field, PrimeField};
     use group::{
-        CurveAffine, CurveProjective, Group, GroupEncoding, PrimeGroup, UncompressedEncoding,
+        CofactorCurve, CurveAffine, Group, GroupEncoding, PrimeGroup, UncompressedEncoding,
     };
     use rand_core::RngCore;
     use std::fmt;
@@ -1462,15 +1458,15 @@ pub mod g1 {
         assert!(b.is_on_curve() && b.is_in_correct_subgroup_assuming_on_curve());
         assert!(c.is_on_curve() && c.is_in_correct_subgroup_assuming_on_curve());
 
-        let mut tmp1 = a.to_projective();
-        tmp1.add_assign(&b.to_projective());
+        let mut tmp1 = a.to_curve();
+        tmp1.add_assign(&b.to_curve());
         assert_eq!(tmp1.to_affine(), c);
-        assert_eq!(tmp1, c.to_projective());
+        assert_eq!(tmp1, c.to_curve());
 
-        let mut tmp2 = a.to_projective();
+        let mut tmp2 = a.to_curve();
         tmp2.add_assign(&b);
         assert_eq!(tmp2.to_affine(), c);
-        assert_eq!(tmp2, c.to_projective());
+        assert_eq!(tmp2, c.to_curve());
     }
 
     #[test]
@@ -1487,7 +1483,7 @@ pub mod g2 {
     use crate::{Engine, PairingCurveAffine};
     use ff::{BitIterator, Field, PrimeField};
     use group::{
-        CurveAffine, CurveProjective, Group, GroupEncoding, PrimeGroup, UncompressedEncoding,
+        CofactorCurve, CurveAffine, Group, GroupEncoding, PrimeGroup, UncompressedEncoding,
     };
     use rand_core::RngCore;
     use std::fmt;
diff --git a/pairing/src/bls12_381/tests/mod.rs b/pairing/src/bls12_381/tests/mod.rs
index e905a486a..1d9f41fc9 100644
--- a/pairing/src/bls12_381/tests/mod.rs
+++ b/pairing/src/bls12_381/tests/mod.rs
@@ -1,5 +1,5 @@
 use ff::PrimeField;
-use group::{CurveAffine, CurveProjective, GroupEncoding, UncompressedEncoding};
+use group::{CofactorCurve, CurveAffine, GroupEncoding, UncompressedEncoding};
 
 use super::*;
 use crate::*;
@@ -55,7 +55,7 @@ fn test_pairing_result_against_relic() {
     });
 }
 
-fn uncompressed_test_vectors<G: CurveProjective>(expected: &[u8])
+fn uncompressed_test_vectors<G: CofactorCurve>(expected: &[u8])
 where
     G::Affine: UncompressedEncoding,
 {
@@ -85,7 +85,7 @@ where
     assert_eq!(&v[..], expected);
 }
 
-fn compressed_test_vectors<G: CurveProjective>(expected: &[u8]) {
+fn compressed_test_vectors<G: CofactorCurve>(expected: &[u8]) {
     let mut e = G::identity();
     let encoded_len = <G::Affine as GroupEncoding>::Repr::default().as_ref().len();
 
diff --git a/pairing/src/lib.rs b/pairing/src/lib.rs
index 6c3e2b47f..6898bd553 100644
--- a/pairing/src/lib.rs
+++ b/pairing/src/lib.rs
@@ -23,7 +23,7 @@ pub mod bls12_381;
 use core::ops::Mul;
 use ff::{Field, PrimeField};
 use group::{
-    CurveAffine, CurveProjective, GroupOps, GroupOpsOwned, ScalarMul, ScalarMulOwned,
+    CofactorCurve, CurveAffine, GroupOps, GroupOpsOwned, ScalarMul, ScalarMulOwned,
     UncompressedEncoding,
 };
 
@@ -35,7 +35,7 @@ pub trait Engine: Sized + 'static + Clone {
     type Fr: PrimeField;
 
     /// The projective representation of an element in G1.
-    type G1: CurveProjective<Scalar = Self::Fr, Affine = Self::G1Affine>
+    type G1: CofactorCurve<Scalar = Self::Fr, Affine = Self::G1Affine>
         + From<Self::G1Affine>
         + GroupOps<Self::G1Affine>
         + GroupOpsOwned<Self::G1Affine>
@@ -45,7 +45,7 @@ pub trait Engine: Sized + 'static + Clone {
     /// The affine representation of an element in G1.
     type G1Affine: PairingCurveAffine<
             Scalar = Self::Fr,
-            Projective = Self::G1,
+            Curve = Self::G1,
             Pair = Self::G2Affine,
             PairingResult = Self::Gt,
         > + From<Self::G1>
@@ -53,7 +53,7 @@ pub trait Engine: Sized + 'static + Clone {
         + for<'a> Mul<&'a Self::Fr, Output = Self::G1>;
 
     /// The projective representation of an element in G2.
-    type G2: CurveProjective<Scalar = Self::Fr, Affine = Self::G2Affine>
+    type G2: CofactorCurve<Scalar = Self::Fr, Affine = Self::G2Affine>
         + From<Self::G2Affine>
         + GroupOps<Self::G2Affine>
         + GroupOpsOwned<Self::G2Affine>
@@ -63,7 +63,7 @@ pub trait Engine: Sized + 'static + Clone {
     /// The affine representation of an element in G2.
     type G2Affine: PairingCurveAffine<
             Scalar = Self::Fr,
-            Projective = Self::G2,
+            Curve = Self::G2,
             Pair = Self::G1Affine,
             PairingResult = Self::Gt,
         > + From<Self::G2>
diff --git a/pairing/src/tests/engine.rs b/pairing/src/tests/engine.rs
index bf0f8849d..2cfc473b2 100644
--- a/pairing/src/tests/engine.rs
+++ b/pairing/src/tests/engine.rs
@@ -1,5 +1,5 @@
 use ff::{Endianness, Field, PrimeField};
-use group::{CurveAffine, CurveProjective, Group};
+use group::{CofactorCurve, CurveAffine, Group};
 use rand_core::SeedableRng;
 use rand_xorshift::XorShiftRng;
 use std::ops::MulAssign;