diff --git a/bellman/src/groth16/prover.rs b/bellman/src/groth16/prover.rs
index 3c5b90f11..c31b4db97 100644
--- a/bellman/src/groth16/prover.rs
+++ b/bellman/src/groth16/prover.rs
@@ -314,34 +314,34 @@ where
     }
 
     let mut g_a = vk.delta_g1.mul(r);
-    g_a.add_assign_mixed(&vk.alpha_g1);
+    AddAssign::<&E::G1Affine>::add_assign(&mut g_a, &vk.alpha_g1);
     let mut g_b = vk.delta_g2.mul(s);
-    g_b.add_assign_mixed(&vk.beta_g2);
+    AddAssign::<&E::G2Affine>::add_assign(&mut g_b, &vk.beta_g2);
     let mut g_c;
     {
         let mut rs = r;
         rs.mul_assign(&s);
 
         g_c = vk.delta_g1.mul(rs);
-        g_c.add_assign(&vk.alpha_g1.mul(s));
-        g_c.add_assign(&vk.beta_g1.mul(r));
+        AddAssign::<&E::G1>::add_assign(&mut g_c, &vk.alpha_g1.mul(s));
+        AddAssign::<&E::G1>::add_assign(&mut g_c, &vk.beta_g1.mul(r));
     }
     let mut a_answer = a_inputs.wait()?;
-    a_answer.add_assign(&a_aux.wait()?);
-    g_a.add_assign(&a_answer);
+    AddAssign::<&E::G1>::add_assign(&mut a_answer, &a_aux.wait()?);
+    AddAssign::<&E::G1>::add_assign(&mut g_a, &a_answer);
     a_answer.mul_assign(s);
-    g_c.add_assign(&a_answer);
+    AddAssign::<&E::G1>::add_assign(&mut g_c, &a_answer);
 
-    let mut b1_answer = b_g1_inputs.wait()?;
-    b1_answer.add_assign(&b_g1_aux.wait()?);
+    let mut b1_answer: E::G1 = b_g1_inputs.wait()?;
+    AddAssign::<&E::G1>::add_assign(&mut b1_answer, &b_g1_aux.wait()?);
     let mut b2_answer = b_g2_inputs.wait()?;
-    b2_answer.add_assign(&b_g2_aux.wait()?);
+    AddAssign::<&E::G2>::add_assign(&mut b2_answer, &b_g2_aux.wait()?);
 
-    g_b.add_assign(&b2_answer);
+    AddAssign::<&E::G2>::add_assign(&mut g_b, &b2_answer);
     b1_answer.mul_assign(r);
-    g_c.add_assign(&b1_answer);
-    g_c.add_assign(&h.wait()?);
-    g_c.add_assign(&l.wait()?);
+    AddAssign::<&E::G1>::add_assign(&mut g_c, &b1_answer);
+    AddAssign::<&E::G1>::add_assign(&mut g_c, &h.wait()?);
+    AddAssign::<&E::G1>::add_assign(&mut g_c, &l.wait()?);
 
     Ok(Proof {
         a: g_a.into_affine(),
diff --git a/bellman/src/groth16/tests/dummy_engine.rs b/bellman/src/groth16/tests/dummy_engine.rs
index 63aca12f9..65a3ba2ca 100644
--- a/bellman/src/groth16/tests/dummy_engine.rs
+++ b/bellman/src/groth16/tests/dummy_engine.rs
@@ -413,18 +413,6 @@ impl CurveProjective for Fr {
         self.0 = <Fr as Field>::double(self).0;
     }
 
-    fn add_assign(&mut self, other: &Self) {
-        AddAssign::add_assign(self, other);
-    }
-
-    fn add_assign_mixed(&mut self, other: &Self) {
-        AddAssign::add_assign(self, other);
-    }
-
-    fn negate(&mut self) {
-        self.0 = self.neg().0;
-    }
-
     fn mul_assign<S: Into<<Self::Scalar as PrimeField>::Repr>>(&mut self, other: S) {
         let tmp = Fr::from_repr(other.into()).unwrap();
 
@@ -503,10 +491,6 @@ impl CurveAffine for Fr {
         <Fr as Field>::is_zero(self)
     }
 
-    fn negate(&mut self) {
-        self.0 = self.neg().0;
-    }
-
     fn mul<S: Into<<Self::Scalar as PrimeField>::Repr>>(&self, other: S) -> Self::Projective {
         let mut res = *self;
         let tmp = Fr::from_repr(other.into()).unwrap();
diff --git a/bellman/src/groth16/verifier.rs b/bellman/src/groth16/verifier.rs
index 5bc05810b..598366715 100644
--- a/bellman/src/groth16/verifier.rs
+++ b/bellman/src/groth16/verifier.rs
@@ -1,16 +1,15 @@
 use ff::PrimeField;
 use group::{CurveAffine, CurveProjective};
 use pairing::{Engine, PairingCurveAffine};
+use std::ops::{AddAssign, Neg};
 
 use super::{PreparedVerifyingKey, Proof, VerifyingKey};
 
 use crate::SynthesisError;
 
 pub fn prepare_verifying_key<E: Engine>(vk: &VerifyingKey<E>) -> PreparedVerifyingKey<E> {
-    let mut gamma = vk.gamma_g2;
-    gamma.negate();
-    let mut delta = vk.delta_g2;
-    delta.negate();
+    let gamma = vk.gamma_g2.neg();
+    let delta = vk.delta_g2.neg();
 
     PreparedVerifyingKey {
         alpha_g1_beta_g2: E::pairing(vk.alpha_g1, vk.beta_g2),
@@ -32,7 +31,7 @@ pub fn verify_proof<'a, E: Engine>(
     let mut acc = pvk.ic[0].into_projective();
 
     for (i, b) in public_inputs.iter().zip(pvk.ic.iter().skip(1)) {
-        acc.add_assign(&b.mul(i.into_repr()));
+        AddAssign::<&E::G1>::add_assign(&mut acc, &b.mul(i.into_repr()));
     }
 
     // The original verification equation is:
diff --git a/bellman/src/multiexp.rs b/bellman/src/multiexp.rs
index b7729a868..0bc61ba14 100644
--- a/bellman/src/multiexp.rs
+++ b/bellman/src/multiexp.rs
@@ -5,6 +5,7 @@ use futures::Future;
 use group::{CurveAffine, CurveProjective};
 use std::io;
 use std::iter;
+use std::ops::AddAssign;
 use std::sync::Arc;
 
 use super::SynthesisError;
@@ -18,16 +19,24 @@ pub trait SourceBuilder<G: CurveAffine>: Send + Sync + 'static + Clone {
 
 /// A source of bases, like an iterator.
 pub trait Source<G: CurveAffine> {
-    /// Parses the element from the source. Fails if the point is at infinity.
-    fn add_assign_mixed(
-        &mut self,
-        to: &mut <G as CurveAffine>::Projective,
-    ) -> Result<(), SynthesisError>;
+    fn next(&mut self) -> Result<&G, SynthesisError>;
 
     /// Skips `amt` elements from the source, avoiding deserialization.
     fn skip(&mut self, amt: usize) -> Result<(), SynthesisError>;
 }
 
+pub trait AddAssignFromSource: CurveProjective {
+    /// Parses the element from the source. Fails if the point is at infinity.
+    fn add_assign_from_source<S: Source<<Self as CurveProjective>::Affine>>(
+        &mut self,
+        source: &mut S,
+    ) -> Result<(), SynthesisError> {
+        AddAssign::<&<Self as CurveProjective>::Affine>::add_assign(self, source.next()?);
+        Ok(())
+    }
+}
+impl<G> AddAssignFromSource for G where G: CurveProjective {}
+
 impl<G: CurveAffine> SourceBuilder<G> for (Arc<Vec<G>>, usize) {
     type Source = (Arc<Vec<G>>, usize);
 
@@ -37,10 +46,7 @@ impl<G: CurveAffine> SourceBuilder<G> for (Arc<Vec<G>>, usize) {
 }
 
 impl<G: CurveAffine> Source<G> for (Arc<Vec<G>>, usize) {
-    fn add_assign_mixed(
-        &mut self,
-        to: &mut <G as CurveAffine>::Projective,
-    ) -> Result<(), SynthesisError> {
+    fn next(&mut self) -> Result<&G, SynthesisError> {
         if self.0.len() <= self.1 {
             return Err(io::Error::new(
                 io::ErrorKind::UnexpectedEof,
@@ -53,11 +59,10 @@ impl<G: CurveAffine> Source<G> for (Arc<Vec<G>>, usize) {
             return Err(SynthesisError::UnexpectedIdentity);
         }
 
-        to.add_assign_mixed(&self.0[self.1]);
-
+        let ret = &self.0[self.1];
         self.1 += 1;
 
-        Ok(())
+        Ok(ret)
     }
 
     fn skip(&mut self, amt: usize) -> Result<(), SynthesisError> {
@@ -153,12 +158,12 @@ fn multiexp_inner<Q, D, G, S>(
     mut skip: u32,
     c: u32,
     handle_trivial: bool,
-) -> Box<dyn Future<Item = <G as CurveAffine>::Projective, Error = SynthesisError>>
+) -> Box<dyn Future<Item = G, Error = SynthesisError>>
 where
     for<'a> &'a Q: QueryDensity,
     D: Send + Sync + 'static + Clone + AsRef<Q>,
-    G: CurveAffine,
-    S: SourceBuilder<G>,
+    G: CurveProjective,
+    S: SourceBuilder<<G as CurveProjective>::Affine>,
 {
     // Perform this region of the multiexp
     let this = {
@@ -168,13 +173,13 @@ where
 
         pool.compute(move || {
             // Accumulate the result
-            let mut acc = G::Projective::zero();
+            let mut acc = G::zero();
 
             // Build a source for the bases
             let mut bases = bases.new();
 
             // Create space for the buckets
-            let mut buckets = vec![<G as CurveAffine>::Projective::zero(); (1 << c) - 1];
+            let mut buckets = vec![G::zero(); (1 << c) - 1];
 
             let zero = <G::Engine as ScalarEngine>::Fr::zero().into_repr();
             let one = <G::Engine as ScalarEngine>::Fr::one().into_repr();
@@ -186,7 +191,7 @@ where
                         bases.skip(1)?;
                     } else if exp == one {
                         if handle_trivial {
-                            bases.add_assign_mixed(&mut acc)?;
+                            acc.add_assign_from_source(&mut bases)?;
                         } else {
                             bases.skip(1)?;
                         }
@@ -196,7 +201,8 @@ where
                         let exp = exp.as_ref()[0] % (1 << c);
 
                         if exp != 0 {
-                            bases.add_assign_mixed(&mut buckets[(exp - 1) as usize])?;
+                            (&mut buckets[(exp - 1) as usize])
+                                .add_assign_from_source(&mut bases)?;
                         } else {
                             bases.skip(1)?;
                         }
@@ -208,7 +214,7 @@ where
             // e.g. 3a + 2b + 1c = a +
             //                    (a) + b +
             //                    ((a) + b) + c
-            let mut running_sum = G::Projective::zero();
+            let mut running_sum = G::zero();
             for exp in buckets.into_iter().rev() {
                 running_sum.add_assign(&exp);
                 acc.add_assign(&running_sum);
@@ -236,7 +242,7 @@ where
                 c,
                 false,
             ))
-            .map(move |(this, mut higher)| {
+            .map(move |(this, mut higher): (_, G)| {
                 for _ in 0..c {
                     higher.double();
                 }
@@ -256,12 +262,12 @@ pub fn multiexp<Q, D, G, S>(
     bases: S,
     density_map: D,
     exponents: Arc<Vec<<<G::Engine as ScalarEngine>::Fr as PrimeField>::Repr>>,
-) -> Box<dyn Future<Item = <G as CurveAffine>::Projective, Error = SynthesisError>>
+) -> Box<dyn Future<Item = G, Error = SynthesisError>>
 where
     for<'a> &'a Q: QueryDensity,
     D: Send + Sync + 'static + Clone + AsRef<Q>,
-    G: CurveAffine,
-    S: SourceBuilder<G>,
+    G: CurveProjective,
+    S: SourceBuilder<<G as CurveProjective>::Affine>,
 {
     let c = if exponents.len() < 32 {
         3u32
@@ -282,16 +288,16 @@ where
 #[cfg(feature = "pairing")]
 #[test]
 fn test_with_bls12() {
-    fn naive_multiexp<G: CurveAffine>(
-        bases: Arc<Vec<G>>,
+    fn naive_multiexp<G: CurveProjective>(
+        bases: Arc<Vec<<G as CurveProjective>::Affine>>,
         exponents: Arc<Vec<<G::Scalar as PrimeField>::Repr>>,
-    ) -> G::Projective {
+    ) -> G {
         assert_eq!(bases.len(), exponents.len());
 
-        let mut acc = G::Projective::zero();
+        let mut acc = G::zero();
 
         for (base, exp) in bases.iter().zip(exponents.iter()) {
-            acc.add_assign(&base.mul(*exp));
+            AddAssign::<&G>::add_assign(&mut acc, &base.mul(*exp));
         }
 
         acc
@@ -314,7 +320,7 @@ fn test_with_bls12() {
             .collect::<Vec<_>>(),
     );
 
-    let naive = naive_multiexp(g.clone(), v.clone());
+    let naive: <Bls12 as Engine>::G1 = naive_multiexp(g.clone(), v.clone());
 
     let pool = Worker::new();
 
diff --git a/group/src/lib.rs b/group/src/lib.rs
index be78b2aef..b4fb22515 100644
--- a/group/src/lib.rs
+++ b/group/src/lib.rs
@@ -5,16 +5,46 @@ use ff::{PrimeField, PrimeFieldDecodingError, ScalarEngine, SqrtField};
 use rand::RngCore;
 use std::error::Error;
 use std::fmt;
+use std::ops::{Add, AddAssign, Neg, Sub, SubAssign};
 
 pub mod tests;
 
 mod wnaf;
 pub use self::wnaf::Wnaf;
 
+/// A helper trait for types implementing group addition.
+pub trait CurveOps<Rhs = Self, Output = Self>:
+    Add<Rhs, Output = Output> + Sub<Rhs, Output = Output> + AddAssign<Rhs> + SubAssign<Rhs>
+{
+}
+
+impl<T, Rhs, Output> CurveOps<Rhs, Output> for T where
+    T: Add<Rhs, Output = Output> + Sub<Rhs, Output = Output> + AddAssign<Rhs> + SubAssign<Rhs>
+{
+}
+
+/// A helper trait for references implementing group addition.
+pub trait CurveOpsOwned<Rhs = Self, Output = Self>: for<'r> CurveOps<&'r Rhs, Output> {}
+impl<T, Rhs, Output> CurveOpsOwned<Rhs, Output> for T where T: for<'r> CurveOps<&'r Rhs, Output> {}
+
 /// Projective representation of an elliptic curve point guaranteed to be
 /// in the correct prime order subgroup.
 pub trait CurveProjective:
-    PartialEq + Eq + Sized + Copy + Clone + Send + Sync + fmt::Debug + fmt::Display + 'static
+    PartialEq
+    + Eq
+    + Sized
+    + Copy
+    + Clone
+    + Send
+    + Sync
+    + fmt::Debug
+    + fmt::Display
+    + 'static
+    + Neg<Output = Self>
+    + CurveOps
+    + CurveOpsOwned
+    + CurveOps<<Self as CurveProjective>::Affine>
+    + CurveOpsOwned<<Self as CurveProjective>::Affine>
 {
     type Engine: ScalarEngine<Fr = Self::Scalar>;
     type Scalar: PrimeField + SqrtField;
@@ -44,22 +74,6 @@ pub trait CurveProjective:
     /// Doubles this element.
     fn double(&mut self);
 
-    /// Adds another element to this element.
-    fn add_assign(&mut self, other: &Self);
-
-    /// Subtracts another element from this element.
-    fn sub_assign(&mut self, other: &Self) {
-        let mut tmp = *other;
-        tmp.negate();
-        self.add_assign(&tmp);
-    }
-
-    /// Adds an affine element to this element.
-    fn add_assign_mixed(&mut self, other: &Self::Affine);
-
-    /// Negates this element.
-    fn negate(&mut self);
-
     /// Performs scalar multiplication of this element.
     fn mul_assign<S: Into<<Self::Scalar as PrimeField>::Repr>>(&mut self, other: S);
 
@@ -78,7 +92,17 @@ pub trait CurveProjective:
 /// Affine representation of an elliptic curve point guaranteed to be
 /// in the correct prime order subgroup.
 pub trait CurveAffine:
-    Copy + Clone + Sized + Send + Sync + fmt::Debug + fmt::Display + PartialEq + Eq + 'static
+    Copy
+    + Clone
+    + Sized
+    + Send
+    + Sync
+    + fmt::Debug
+    + fmt::Display
+    + PartialEq
+    + Eq
+    + 'static
+    + Neg<Output = Self>
 {
     type Engine: ScalarEngine<Fr = Self::Scalar>;
     type Scalar: PrimeField + SqrtField;
@@ -97,9 +121,6 @@ pub trait CurveAffine:
     /// additive identity.
     fn is_zero(&self) -> bool;
 
-    /// Negates this element.
-    fn negate(&mut self);
-
     /// Performs scalar multiplication of this element with mixed addition.
     fn mul<S: Into<<Self::Scalar as PrimeField>::Repr>>(&self, other: S) -> Self::Projective;
 
diff --git a/group/src/tests/mod.rs b/group/src/tests/mod.rs
index 949d934f2..c53ba762f 100644
--- a/group/src/tests/mod.rs
+++ b/group/src/tests/mod.rs
@@ -13,8 +13,7 @@ pub fn curve_tests<G: CurveProjective>() {
 
     // Negation edge case with zero.
     {
-        let mut z = G::zero();
-        z.negate();
+        let z = G::zero().neg();
         assert!(z.is_zero());
     }
 
@@ -31,19 +30,19 @@ pub fn curve_tests<G: CurveProjective>() {
         let rcopy = r;
         r.add_assign(&G::zero());
         assert_eq!(r, rcopy);
-        r.add_assign_mixed(&G::Affine::zero());
+        r.add_assign(&G::Affine::zero());
         assert_eq!(r, rcopy);
 
         let mut z = G::zero();
         z.add_assign(&G::zero());
         assert!(z.is_zero());
-        z.add_assign_mixed(&G::Affine::zero());
+        z.add_assign(&G::Affine::zero());
         assert!(z.is_zero());
 
         let mut z2 = z;
         z2.add_assign(&r);
 
-        z.add_assign_mixed(&r.into_affine());
+        z.add_assign(&r.into_affine());
 
         assert_eq!(z, z2);
         assert_eq!(z, r);
@@ -68,7 +67,7 @@ pub fn curve_tests<G: CurveProjective>() {
     random_negation_tests::<G>();
     random_transformation_tests::<G>();
     random_wnaf_tests::<G>();
-    random_encoding_tests::<G::Affine>();
+    random_encoding_tests::<G>();
 }
 
 fn random_wnaf_tests<G: CurveProjective>() {
@@ -213,11 +212,10 @@ fn random_negation_tests<G: CurveProjective>() {
         assert!(t3.is_zero());
 
         let mut t4 = t1;
-        t4.add_assign_mixed(&t2.into_affine());
+        t4.add_assign(&t2.into_affine());
         assert!(t4.is_zero());
 
-        t1.negate();
-        assert_eq!(t1, t2);
+        assert_eq!(t1.neg(), t2);
     }
 }
 
@@ -244,7 +242,7 @@ fn random_doubling_tests<G: CurveProjective>() {
         tmp2.add_assign(&b);
 
         let mut tmp3 = a;
-        tmp3.add_assign_mixed(&b.into_affine());
+        tmp3.add_assign(&b.into_affine());
 
         assert_eq!(tmp1, tmp2);
         assert_eq!(tmp1, tmp3);
@@ -306,7 +304,7 @@ fn random_addition_tests<G: CurveProjective>() {
             aplusa.add_assign(&a);
 
             let mut aplusamixed = a;
-            aplusamixed.add_assign_mixed(&a.into_affine());
+            aplusamixed.add_assign(&a.into_affine());
 
             let mut adouble = a;
             adouble.double();
@@ -336,18 +334,18 @@ fn random_addition_tests<G: CurveProjective>() {
 
         // (a + b) + c
         tmp[3] = a_affine.into_projective();
-        tmp[3].add_assign_mixed(&b_affine);
-        tmp[3].add_assign_mixed(&c_affine);
+        tmp[3].add_assign(&b_affine);
+        tmp[3].add_assign(&c_affine);
 
         // a + (b + c)
         tmp[4] = b_affine.into_projective();
-        tmp[4].add_assign_mixed(&c_affine);
-        tmp[4].add_assign_mixed(&a_affine);
+        tmp[4].add_assign(&c_affine);
+        tmp[4].add_assign(&a_affine);
 
         // (a + c) + b
         tmp[5] = a_affine.into_projective();
-        tmp[5].add_assign_mixed(&c_affine);
-        tmp[5].add_assign_mixed(&b_affine);
+        tmp[5].add_assign(&c_affine);
+        tmp[5].add_assign(&b_affine);
 
         // Comparisons
         for i in 0..6 {
@@ -413,24 +411,24 @@ fn random_transformation_tests<G: CurveProjective>() {
     }
 }
 
-fn random_encoding_tests<G: CurveAffine>() {
+fn random_encoding_tests<G: CurveProjective>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
     ]);
 
     assert_eq!(
-        G::zero().into_uncompressed().into_affine().unwrap(),
-        G::zero()
+        G::Affine::zero().into_uncompressed().into_affine().unwrap(),
+        G::Affine::zero()
     );
 
     assert_eq!(
-        G::zero().into_compressed().into_affine().unwrap(),
-        G::zero()
+        G::Affine::zero().into_compressed().into_affine().unwrap(),
+        G::Affine::zero()
     );
 
     for _ in 0..1000 {
-        let mut r = G::Projective::random(&mut rng).into_affine();
+        let mut r = G::random(&mut rng).into_affine();
 
         let uncompressed = r.into_uncompressed();
         let de_uncompressed = uncompressed.into_affine().unwrap();
@@ -440,7 +438,7 @@ fn random_encoding_tests<G: CurveAffine>() {
         let de_compressed = compressed.into_affine().unwrap();
         assert_eq!(de_compressed, r);
 
-        r.negate();
+        r = r.neg();
 
         let compressed = r.into_compressed();
         let de_compressed = compressed.into_affine().unwrap();
diff --git a/pairing/benches/bls12_381/ec.rs b/pairing/benches/bls12_381/ec.rs
index 34113629d..28ce4bf84 100644
--- a/pairing/benches/bls12_381/ec.rs
+++ b/pairing/benches/bls12_381/ec.rs
@@ -2,6 +2,7 @@ pub(crate) mod g1 {
     use criterion::{criterion_group, Criterion};
     use rand_core::SeedableRng;
     use rand_xorshift::XorShiftRng;
+    use std::ops::AddAssign;
 
     use ff::Field;
     use group::CurveProjective;
@@ -69,7 +70,7 @@ pub(crate) mod g1 {
         c.bench_function("G1::add_assign_mixed", |b| {
             b.iter(|| {
                 let mut tmp = v[count].0;
-                tmp.add_assign_mixed(&v[count].1);
+                tmp.add_assign(&v[count].1);
                 count = (count + 1) % SAMPLES;
                 tmp
             })
@@ -88,6 +89,7 @@ pub(crate) mod g2 {
     use criterion::{criterion_group, Criterion};
     use rand_core::SeedableRng;
     use rand_xorshift::XorShiftRng;
+    use std::ops::AddAssign;
 
     use ff::Field;
     use group::CurveProjective;
@@ -155,7 +157,7 @@ pub(crate) mod g2 {
         c.bench_function("G2::add_assign_mixed", |b| {
             b.iter(|| {
                 let mut tmp = v[count].0;
-                tmp.add_assign_mixed(&v[count].1);
+                tmp.add_assign(&v[count].1);
                 count = (count + 1) % SAMPLES;
                 tmp
             })
diff --git a/pairing/src/bls12_381/ec.rs b/pairing/src/bls12_381/ec.rs
index 513228934..8f8187fd4 100644
--- a/pairing/src/bls12_381/ec.rs
+++ b/pairing/src/bls12_381/ec.rs
@@ -86,7 +86,7 @@ macro_rules! curve_impl {
                 for i in bits {
                     res.double();
                     if i {
-                        res.add_assign_mixed(self)
+                        res.add_assign(self)
                     }
                 }
                 res
@@ -134,6 +134,19 @@ macro_rules! curve_impl {
             }
         }
 
+        impl ::std::ops::Neg for $affine {
+            type Output = Self;
+
+            #[inline]
+            fn neg(self) -> Self {
+                let mut ret = self;
+                if !ret.is_zero() {
+                    ret.y = ret.y.neg();
+                }
+                ret
+            }
+        }
+
         impl CurveAffine for $affine {
             type Engine = Bls12;
             type Scalar = $scalarfield;
@@ -163,12 +176,6 @@ macro_rules! curve_impl {
                 self.mul_bits(bits)
             }
 
-            fn negate(&mut self) {
-                if !self.is_zero() {
-                    self.y = self.y.neg();
-                }
-            }
-
             fn into_projective(&self) -> $projective {
                 (*self).into()
             }
@@ -188,6 +195,309 @@ macro_rules! curve_impl {
             }
         }
 
+        impl ::std::ops::Neg for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn neg(self) -> Self {
+                let mut ret = self;
+                if !ret.is_zero() {
+                    ret.y = ret.y.neg();
+                }
+                ret
+            }
+        }
+
+        impl<'r> ::std::ops::Add<&'r $projective> for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn add(self, other: &Self) -> Self {
+                let mut ret = self;
+                ret.add_assign(other);
+                ret
+            }
+        }
+
+        impl ::std::ops::Add for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn add(self, other: Self) -> Self {
+                self + &other
+            }
+        }
+
+        impl<'r> ::std::ops::AddAssign<&'r $projective> for $projective {
+            fn add_assign(&mut self, other: &Self) {
+                if self.is_zero() {
+                    *self = *other;
+                    return;
+                }
+
+                if other.is_zero() {
+                    return;
+                }
+
+                // http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
+
+                // Z1Z1 = Z1^2
+                let z1z1 = self.z.square();
+
+                // Z2Z2 = Z2^2
+                let z2z2 = other.z.square();
+
+                // U1 = X1*Z2Z2
+                let mut u1 = self.x;
+                u1.mul_assign(&z2z2);
+
+                // U2 = X2*Z1Z1
+                let mut u2 = other.x;
+                u2.mul_assign(&z1z1);
+
+                // S1 = Y1*Z2*Z2Z2
+                let mut s1 = self.y;
+                s1.mul_assign(&other.z);
+                s1.mul_assign(&z2z2);
+
+                // S2 = Y2*Z1*Z1Z1
+                let mut s2 = other.y;
+                s2.mul_assign(&self.z);
+                s2.mul_assign(&z1z1);
+
+                if u1 == u2 && s1 == s2 {
+                    // The two points are equal, so we double.
+                    self.double();
+                } else {
+                    // If we're adding -a and a together, self.z becomes zero as H becomes zero.
+
+                    // H = U2-U1
+                    let mut h = u2;
+                    h.sub_assign(&u1);
+
+                    // I = (2*H)^2
+                    let i = h.double().square();
+
+                    // J = H*I
+                    let mut j = h;
+                    j.mul_assign(&i);
+
+                    // r = 2*(S2-S1)
+                    let mut r = s2;
+                    r.sub_assign(&s1);
+                    r = r.double();
+
+                    // V = U1*I
+                    let mut v = u1;
+                    v.mul_assign(&i);
+
+                    // X3 = r^2 - J - 2*V
+                    self.x = r.square();
+                    self.x.sub_assign(&j);
+                    self.x.sub_assign(&v);
+                    self.x.sub_assign(&v);
+
+                    // Y3 = r*(V - X3) - 2*S1*J
+                    self.y = v;
+                    self.y.sub_assign(&self.x);
+                    self.y.mul_assign(&r);
+                    s1.mul_assign(&j); // S1 = S1 * J * 2
+                    s1 = s1.double();
+                    self.y.sub_assign(&s1);
+
+                    // Z3 = ((Z1+Z2)^2 - Z1Z1 - Z2Z2)*H
+                    self.z.add_assign(&other.z);
+                    self.z = self.z.square();
+                    self.z.sub_assign(&z1z1);
+                    self.z.sub_assign(&z2z2);
+                    self.z.mul_assign(&h);
+                }
+            }
+        }
+
+        impl ::std::ops::AddAssign for $projective {
+            #[inline]
+            fn add_assign(&mut self, other: Self) {
+                self.add_assign(&other);
+            }
+        }
+
+        impl<'r> ::std::ops::Sub<&'r $projective> for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn sub(self, other: &Self) -> Self {
+                let mut ret = self;
+                ret.sub_assign(other);
+                ret
+            }
+        }
+
+        impl ::std::ops::Sub for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn sub(self, other: Self) -> Self {
+                self - &other
+            }
+        }
+
+        impl<'r> ::std::ops::SubAssign<&'r $projective> for $projective {
+            fn sub_assign(&mut self, other: &Self) {
+                self.add_assign(&other.neg());
+            }
+        }
+
+        impl ::std::ops::SubAssign for $projective {
+            #[inline]
+            fn sub_assign(&mut self, other: Self) {
+                self.sub_assign(&other);
+            }
+        }
+
+        impl<'r> ::std::ops::Add<&'r <$projective as CurveProjective>::Affine> for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn add(self, other: &<$projective as CurveProjective>::Affine) -> Self {
+                let mut ret = self;
+                ret.add_assign(other);
+                ret
+            }
+        }
+
+        impl ::std::ops::Add<<$projective as CurveProjective>::Affine> for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn add(self, other: <$projective as CurveProjective>::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) {
+                if other.is_zero() {
+                    return;
+                }
+
+                if self.is_zero() {
+                    self.x = other.x;
+                    self.y = other.y;
+                    self.z = $basefield::one();
+                    return;
+                }
+
+                // http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
+
+                // Z1Z1 = Z1^2
+                let z1z1 = self.z.square();
+
+                // U2 = X2*Z1Z1
+                let mut u2 = other.x;
+                u2.mul_assign(&z1z1);
+
+                // S2 = Y2*Z1*Z1Z1
+                let mut s2 = other.y;
+                s2.mul_assign(&self.z);
+                s2.mul_assign(&z1z1);
+
+                if self.x == u2 && self.y == s2 {
+                    // The two points are equal, so we double.
+                    self.double();
+                } else {
+                    // If we're adding -a and a together, self.z becomes zero as H becomes zero.
+
+                    // H = U2-X1
+                    let mut h = u2;
+                    h.sub_assign(&self.x);
+
+                    // HH = H^2
+                    let hh = h.square();
+
+                    // I = 4*HH
+                    let i = hh.double().double();
+
+                    // J = H*I
+                    let mut j = h;
+                    j.mul_assign(&i);
+
+                    // r = 2*(S2-Y1)
+                    let mut r = s2;
+                    r.sub_assign(&self.y);
+                    r = r.double();
+
+                    // V = X1*I
+                    let mut v = self.x;
+                    v.mul_assign(&i);
+
+                    // X3 = r^2 - J - 2*V
+                    self.x = r.square();
+                    self.x.sub_assign(&j);
+                    self.x.sub_assign(&v);
+                    self.x.sub_assign(&v);
+
+                    // Y3 = r*(V-X3)-2*Y1*J
+                    j.mul_assign(&self.y); // J = 2*Y1*J
+                    j = j.double();
+                    self.y = v;
+                    self.y.sub_assign(&self.x);
+                    self.y.mul_assign(&r);
+                    self.y.sub_assign(&j);
+
+                    // Z3 = (Z1+H)^2-Z1Z1-HH
+                    self.z.add_assign(&h);
+                    self.z = self.z.square();
+                    self.z.sub_assign(&z1z1);
+                    self.z.sub_assign(&hh);
+                }
+            }
+        }
+
+        impl ::std::ops::AddAssign<<$projective as CurveProjective>::Affine> for $projective {
+            #[inline]
+            fn add_assign(&mut self, other: <$projective as CurveProjective>::Affine) {
+                self.add_assign(&other);
+            }
+        }
+
+        impl<'r> ::std::ops::Sub<&'r <$projective as CurveProjective>::Affine> for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn sub(self, other: &<$projective as CurveProjective>::Affine) -> Self {
+                let mut ret = self;
+                ret.sub_assign(other);
+                ret
+            }
+        }
+
+        impl ::std::ops::Sub<<$projective as CurveProjective>::Affine> for $projective {
+            type Output = Self;
+
+            #[inline]
+            fn sub(self, other: <$projective as CurveProjective>::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) {
+                self.add_assign(&other.neg());
+            }
+        }
+
+        impl ::std::ops::SubAssign<<$projective as CurveProjective>::Affine> for $projective {
+            #[inline]
+            fn sub_assign(&mut self, other: <$projective as CurveProjective>::Affine) {
+                self.sub_assign(&other);
+            }
+        }
+
         impl CurveProjective for $projective {
             type Engine = Bls12;
             type Scalar = $scalarfield;
@@ -340,174 +650,6 @@ macro_rules! curve_impl {
                 self.y.sub_assign(&c);
             }
 
-            fn add_assign(&mut self, other: &Self) {
-                if self.is_zero() {
-                    *self = *other;
-                    return;
-                }
-
-                if other.is_zero() {
-                    return;
-                }
-
-                // http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
-
-                // Z1Z1 = Z1^2
-                let z1z1 = self.z.square();
-
-                // Z2Z2 = Z2^2
-                let z2z2 = other.z.square();
-
-                // U1 = X1*Z2Z2
-                let mut u1 = self.x;
-                u1.mul_assign(&z2z2);
-
-                // U2 = X2*Z1Z1
-                let mut u2 = other.x;
-                u2.mul_assign(&z1z1);
-
-                // S1 = Y1*Z2*Z2Z2
-                let mut s1 = self.y;
-                s1.mul_assign(&other.z);
-                s1.mul_assign(&z2z2);
-
-                // S2 = Y2*Z1*Z1Z1
-                let mut s2 = other.y;
-                s2.mul_assign(&self.z);
-                s2.mul_assign(&z1z1);
-
-                if u1 == u2 && s1 == s2 {
-                    // The two points are equal, so we double.
-                    self.double();
-                } else {
-                    // If we're adding -a and a together, self.z becomes zero as H becomes zero.
-
-                    // H = U2-U1
-                    let mut h = u2;
-                    h.sub_assign(&u1);
-
-                    // I = (2*H)^2
-                    let i = h.double().square();
-
-                    // J = H*I
-                    let mut j = h;
-                    j.mul_assign(&i);
-
-                    // r = 2*(S2-S1)
-                    let mut r = s2;
-                    r.sub_assign(&s1);
-                    r = r.double();
-
-                    // V = U1*I
-                    let mut v = u1;
-                    v.mul_assign(&i);
-
-                    // X3 = r^2 - J - 2*V
-                    self.x = r.square();
-                    self.x.sub_assign(&j);
-                    self.x.sub_assign(&v);
-                    self.x.sub_assign(&v);
-
-                    // Y3 = r*(V - X3) - 2*S1*J
-                    self.y = v;
-                    self.y.sub_assign(&self.x);
-                    self.y.mul_assign(&r);
-                    s1.mul_assign(&j); // S1 = S1 * J * 2
-                    s1 = s1.double();
-                    self.y.sub_assign(&s1);
-
-                    // Z3 = ((Z1+Z2)^2 - Z1Z1 - Z2Z2)*H
-                    self.z.add_assign(&other.z);
-                    self.z = self.z.square();
-                    self.z.sub_assign(&z1z1);
-                    self.z.sub_assign(&z2z2);
-                    self.z.mul_assign(&h);
-                }
-            }
-
-            fn add_assign_mixed(&mut self, other: &Self::Affine) {
-                if other.is_zero() {
-                    return;
-                }
-
-                if self.is_zero() {
-                    self.x = other.x;
-                    self.y = other.y;
-                    self.z = $basefield::one();
-                    return;
-                }
-
-                // http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
-
-                // Z1Z1 = Z1^2
-                let z1z1 = self.z.square();
-
-                // U2 = X2*Z1Z1
-                let mut u2 = other.x;
-                u2.mul_assign(&z1z1);
-
-                // S2 = Y2*Z1*Z1Z1
-                let mut s2 = other.y;
-                s2.mul_assign(&self.z);
-                s2.mul_assign(&z1z1);
-
-                if self.x == u2 && self.y == s2 {
-                    // The two points are equal, so we double.
-                    self.double();
-                } else {
-                    // If we're adding -a and a together, self.z becomes zero as H becomes zero.
-
-                    // H = U2-X1
-                    let mut h = u2;
-                    h.sub_assign(&self.x);
-
-                    // HH = H^2
-                    let hh = h.square();
-
-                    // I = 4*HH
-                    let i = hh.double().double();
-
-                    // J = H*I
-                    let mut j = h;
-                    j.mul_assign(&i);
-
-                    // r = 2*(S2-Y1)
-                    let mut r = s2;
-                    r.sub_assign(&self.y);
-                    r = r.double();
-
-                    // V = X1*I
-                    let mut v = self.x;
-                    v.mul_assign(&i);
-
-                    // X3 = r^2 - J - 2*V
-                    self.x = r.square();
-                    self.x.sub_assign(&j);
-                    self.x.sub_assign(&v);
-                    self.x.sub_assign(&v);
-
-                    // Y3 = r*(V-X3)-2*Y1*J
-                    j.mul_assign(&self.y); // J = 2*Y1*J
-                    j = j.double();
-                    self.y = v;
-                    self.y.sub_assign(&self.x);
-                    self.y.mul_assign(&r);
-                    self.y.sub_assign(&j);
-
-                    // Z3 = (Z1+H)^2-Z1Z1-HH
-                    self.z.add_assign(&h);
-                    self.z = self.z.square();
-                    self.z.sub_assign(&z1z1);
-                    self.z.sub_assign(&hh);
-                }
-            }
-
-            fn negate(&mut self) {
-                if !self.is_zero() {
-                    self.y = self.y.neg();
-                }
-            }
-
             fn mul_assign<S: Into<<Self::Scalar as PrimeField>::Repr>>(&mut self, other: S) {
                 let mut res = Self::zero();
 
@@ -521,7 +663,7 @@ macro_rules! curve_impl {
                     }
 
                     if i {
-                        res.add_assign(self);
+                        res.add_assign(&*self);
                     }
                 }
 
@@ -1258,7 +1400,7 @@ pub mod g1 {
         assert_eq!(tmp1, c.into_projective());
 
         let mut tmp2 = a.into_projective();
-        tmp2.add_assign_mixed(&b);
+        tmp2.add_assign(&b);
         assert_eq!(tmp2.into_affine(), c);
         assert_eq!(tmp2, c.into_projective());
     }
diff --git a/pairing/src/lib.rs b/pairing/src/lib.rs
index bd060a12f..0081d811e 100644
--- a/pairing/src/lib.rs
+++ b/pairing/src/lib.rs
@@ -21,7 +21,7 @@ pub mod tests;
 pub mod bls12_381;
 
 use ff::{Field, PrimeField, ScalarEngine, SqrtField};
-use group::{CurveAffine, CurveProjective};
+use group::{CurveAffine, CurveOps, CurveOpsOwned, CurveProjective};
 use subtle::CtOption;
 
 /// An "engine" is a collection of types (fields, elliptic curve groups, etc.)
@@ -30,7 +30,9 @@ use subtle::CtOption;
 pub trait Engine: ScalarEngine {
     /// The projective representation of an element in G1.
     type G1: CurveProjective<Engine = Self, Base = Self::Fq, Scalar = Self::Fr, Affine = Self::G1Affine>
-        + From<Self::G1Affine>;
+        + From<Self::G1Affine>
+        + CurveOps<Self::G1Affine>
+        + CurveOpsOwned<Self::G1Affine>;
 
     /// The affine representation of an element in G1.
     type G1Affine: PairingCurveAffine<
@@ -44,7 +46,9 @@ pub trait Engine: ScalarEngine {
 
     /// The projective representation of an element in G2.
     type G2: CurveProjective<Engine = Self, Base = Self::Fqe, Scalar = Self::Fr, Affine = Self::G2Affine>
-        + From<Self::G2Affine>;
+        + From<Self::G2Affine>
+        + CurveOps<Self::G2Affine>
+        + CurveOpsOwned<Self::G2Affine>;
 
     /// The affine representation of an element in G2.
     type G2Affine: PairingCurveAffine<