diff --git a/bellman/src/groth16/tests/dummy_engine.rs b/bellman/src/groth16/tests/dummy_engine.rs
index f75be5369..504030fd7 100644
--- a/bellman/src/groth16/tests/dummy_engine.rs
+++ b/bellman/src/groth16/tests/dummy_engine.rs
@@ -417,10 +417,6 @@ impl CurveProjective for Fr {
         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();
 
@@ -499,10 +495,6 @@ 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 3d9ff0881..8ee8a7418 100644
--- a/bellman/src/groth16/verifier.rs
+++ b/bellman/src/groth16/verifier.rs
@@ -1,17 +1,15 @@
 use ff::PrimeField;
 use group::{CurveAffine, CurveProjective};
 use pairing::{Engine, PairingCurveAffine};
-use std::ops::AddAssign;
+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),
diff --git a/group/src/lib.rs b/group/src/lib.rs
index 8d5c405a6..b2dc410ea 100644
--- a/group/src/lib.rs
+++ b/group/src/lib.rs
@@ -5,7 +5,7 @@ use ff::{PrimeField, PrimeFieldDecodingError, ScalarEngine, SqrtField};
 use rand::RngCore;
 use std::error::Error;
 use std::fmt;
-use std::ops::{Add, AddAssign, Sub, SubAssign};
+use std::ops::{Add, AddAssign, Neg, Sub, SubAssign};
 
 pub mod tests;
 
@@ -27,6 +27,7 @@ pub trait CurveProjective:
     + 'static
     + Add<Output = Self>
     + Sub<Output = Self>
+    + Neg<Output = Self>
     + for<'a> Add<&'a Self, Output = Self>
     + for<'a> Sub<&'a Self, Output = Self>
     + AddAssign
@@ -65,9 +66,6 @@ pub trait CurveProjective:
     /// 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);
 
@@ -86,7 +84,17 @@ 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;
@@ -105,9 +113,6 @@ 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..e86167b41 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());
     }
 
@@ -216,8 +215,7 @@ fn random_negation_tests<G: CurveProjective>() {
         t4.add_assign_mixed(&t2.into_affine());
         assert!(t4.is_zero());
 
-        t1.negate();
-        assert_eq!(t1, t2);
+        assert_eq!(t1.neg(), t2);
     }
 }
 
@@ -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/src/bls12_381/ec.rs b/pairing/src/bls12_381/ec.rs
index 8e105f290..b016959d7 100644
--- a/pairing/src/bls12_381/ec.rs
+++ b/pairing/src/bls12_381/ec.rs
@@ -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 +195,19 @@ 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;
 
@@ -324,9 +344,7 @@ macro_rules! curve_impl {
 
         impl<'r> ::std::ops::SubAssign<&'r $projective> for $projective {
             fn sub_assign(&mut self, other: &Self) {
-                let mut tmp = *other;
-                tmp.negate();
-                self.add_assign(&tmp);
+                self.add_assign(&other.neg());
             }
         }
 
@@ -566,12 +584,6 @@ macro_rules! curve_impl {
                 }
             }
 
-            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();