diff --git a/bellman/src/gadgets/lookup.rs b/bellman/src/gadgets/lookup.rs
index 3be3ed957..bde86e2ba 100644
--- a/bellman/src/gadgets/lookup.rs
+++ b/bellman/src/gadgets/lookup.rs
@@ -1,7 +1,7 @@
 //! Window table lookup gadgets.
 
 use ff::{Field, ScalarEngine};
-use std::ops::AddAssign;
+use std::ops::{AddAssign, Neg};
 
 use super::boolean::Boolean;
 use super::num::{AllocatedNum, Num};
@@ -16,8 +16,7 @@ where
     assert_eq!(assignment.len(), 1 << window_size);
 
     for (i, constant) in constants.into_iter().enumerate() {
-        let mut cur = assignment[i];
-        cur.negate();
+        let mut cur = assignment[i].neg();
         cur.add_assign(constant);
         assignment[i] = cur;
         for (j, eval) in assignment.iter_mut().enumerate().skip(i + 1) {
@@ -151,7 +150,7 @@ where
     let y = AllocatedNum::alloc(cs.namespace(|| "y"), || {
         let mut tmp = coords[*i.get()?].1;
         if *bits[2].get_value().get()? {
-            tmp.negate();
+            tmp = tmp.neg();
         }
         Ok(tmp)
     })?;
@@ -281,7 +280,7 @@ mod test {
             assert_eq!(res.0.get_value().unwrap(), points[index].0);
             let mut tmp = points[index].1;
             if c_val {
-                tmp.negate()
+                tmp = tmp.neg()
             }
             assert_eq!(res.1.get_value().unwrap(), tmp);
         }
diff --git a/bellman/src/gadgets/num.rs b/bellman/src/gadgets/num.rs
index da3e4a060..bce55ced1 100644
--- a/bellman/src/gadgets/num.rs
+++ b/bellman/src/gadgets/num.rs
@@ -417,7 +417,7 @@ mod test {
     use pairing::bls12_381::{Bls12, Fr};
     use rand_core::SeedableRng;
     use rand_xorshift::XorShiftRng;
-    use std::ops::SubAssign;
+    use std::ops::{Neg, SubAssign};
 
     use super::{AllocatedNum, Boolean};
     use crate::gadgets::test::*;
@@ -519,8 +519,7 @@ mod test {
 
     #[test]
     fn test_into_bits_strict() {
-        let mut negone = Fr::one();
-        negone.negate();
+        let negone = Fr::one().neg();
 
         let mut cs = TestConstraintSystem::<Bls12>::new();
 
diff --git a/bellman/src/gadgets/test/mod.rs b/bellman/src/gadgets/test/mod.rs
index f0668b4a2..f4cc92744 100644
--- a/bellman/src/gadgets/test/mod.rs
+++ b/bellman/src/gadgets/test/mod.rs
@@ -6,7 +6,7 @@ use crate::{ConstraintSystem, Index, LinearCombination, SynthesisError, Variable
 
 use std::collections::HashMap;
 use std::fmt::Write;
-use std::ops::{AddAssign, MulAssign};
+use std::ops::{AddAssign, MulAssign, Neg};
 
 use byteorder::{BigEndian, ByteOrder};
 use std::cmp::Ordering;
@@ -152,11 +152,7 @@ impl<E: ScalarEngine> TestConstraintSystem<E> {
     pub fn pretty_print(&self) -> String {
         let mut s = String::new();
 
-        let negone = {
-            let mut tmp = E::Fr::one();
-            tmp.negate();
-            tmp
-        };
+        let negone = E::Fr::one().neg();
 
         let powers_of_two = (0..E::Fr::NUM_BITS)
             .map(|i| E::Fr::from_str("2").unwrap().pow(&[u64::from(i)]))
diff --git a/bellman/src/groth16/tests/dummy_engine.rs b/bellman/src/groth16/tests/dummy_engine.rs
index 325c198d2..46641b347 100644
--- a/bellman/src/groth16/tests/dummy_engine.rs
+++ b/bellman/src/groth16/tests/dummy_engine.rs
@@ -9,7 +9,7 @@ use rand_core::RngCore;
 use std::cmp::Ordering;
 use std::fmt;
 use std::num::Wrapping;
-use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};
+use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
 const MODULUS_R: Wrapping<u32> = Wrapping(64513);
 
@@ -22,6 +22,17 @@ impl fmt::Display for Fr {
     }
 }
 
+impl Neg for Fr {
+    type Output = Self;
+
+    fn neg(mut self) -> Self {
+        if !<Fr as Field>::is_zero(&self) {
+            self.0 = MODULUS_R - self.0;
+        }
+        self
+    }
+}
+
 impl<'r> Add<&'r Fr> for Fr {
     type Output = Self;
 
@@ -137,12 +148,6 @@ impl Field for Fr {
         self.0 = (self.0 << 1) % MODULUS_R;
     }
 
-    fn negate(&mut self) {
-        if !<Fr as Field>::is_zero(self) {
-            self.0 = MODULUS_R - self.0;
-        }
-    }
-
     fn inverse(&self) -> Option<Self> {
         if <Fr as Field>::is_zero(self) {
             None
@@ -413,7 +418,7 @@ impl CurveProjective for Fr {
     }
 
     fn negate(&mut self) {
-        <Fr as Field>::negate(self);
+        self.0 = self.neg().0;
     }
 
     fn mul_assign<S: Into<<Self::Scalar as PrimeField>::Repr>>(&mut self, other: S) {
@@ -495,7 +500,7 @@ impl CurveAffine for Fr {
     }
 
     fn negate(&mut self) {
-        <Fr as Field>::negate(self);
+        self.0 = self.neg().0;
     }
 
     fn mul<S: Into<<Self::Scalar as PrimeField>::Repr>>(&self, other: S) -> Self::Projective {
diff --git a/bellman/src/lib.rs b/bellman/src/lib.rs
index 3877c3f4d..1e48b0c79 100644
--- a/bellman/src/lib.rs
+++ b/bellman/src/lib.rs
@@ -148,7 +148,7 @@ use std::error::Error;
 use std::fmt;
 use std::io;
 use std::marker::PhantomData;
-use std::ops::{Add, MulAssign, Sub};
+use std::ops::{Add, MulAssign, Neg, Sub};
 
 /// Computations are expressed in terms of arithmetic circuits, in particular
 /// rank-1 quadratic constraint systems. The `Circuit` trait represents a
@@ -216,10 +216,8 @@ impl<E: ScalarEngine> Sub<(E::Fr, Variable)> for LinearCombination<E> {
     type Output = LinearCombination<E>;
 
     #[allow(clippy::suspicious_arithmetic_impl)]
-    fn sub(self, (mut coeff, var): (E::Fr, Variable)) -> LinearCombination<E> {
-        coeff.negate();
-
-        self + (coeff, var)
+    fn sub(self, (coeff, var): (E::Fr, Variable)) -> LinearCombination<E> {
+        self + (coeff.neg(), var)
     }
 }
 
diff --git a/ff/ff_derive/src/lib.rs b/ff/ff_derive/src/lib.rs
index b230e7f3e..0e804c786 100644
--- a/ff/ff_derive/src/lib.rs
+++ b/ff/ff_derive/src/lib.rs
@@ -833,6 +833,21 @@ fn prime_field_impl(
             }
         }
 
+        impl ::std::ops::Neg for #name {
+            type Output = #name;
+
+            #[inline]
+            fn neg(self) -> #name {
+                let mut ret = self;
+                if !ret.is_zero() {
+                    let mut tmp = MODULUS;
+                    tmp.sub_noborrow(&ret.0);
+                    ret.0 = tmp;
+                }
+                ret
+            }
+        }
+
         impl<'r> ::std::ops::Add<&'r #name> for #name {
             type Output = #name;
 
@@ -1033,15 +1048,6 @@ fn prime_field_impl(
                 self.reduce();
             }
 
-            #[inline]
-            fn negate(&mut self) {
-                if !self.is_zero() {
-                    let mut tmp = MODULUS;
-                    tmp.sub_noborrow(&self.0);
-                    self.0 = tmp;
-                }
-            }
-
             fn inverse(&self) -> Option<Self> {
                 if self.is_zero() {
                     None
diff --git a/ff/src/lib.rs b/ff/src/lib.rs
index 71a58704e..c605f6433 100644
--- a/ff/src/lib.rs
+++ b/ff/src/lib.rs
@@ -11,7 +11,7 @@ use rand_core::RngCore;
 use std::error::Error;
 use std::fmt;
 use std::io::{self, Read, Write};
-use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};
+use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
 /// This trait represents an element of a field.
 pub trait Field:
@@ -27,6 +27,7 @@ pub trait Field:
     + 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>
@@ -55,9 +56,6 @@ pub trait Field:
     /// Doubles this element.
     fn double(&mut self);
 
-    /// Negates this element.
-    fn negate(&mut self);
-
     /// Computes the multiplicative inverse of this element, if nonzero.
     fn inverse(&self) -> Option<Self>;
 
diff --git a/group/src/tests/mod.rs b/group/src/tests/mod.rs
index 1970667f8..949d934f2 100644
--- a/group/src/tests/mod.rs
+++ b/group/src/tests/mod.rs
@@ -1,6 +1,7 @@
 use ff::{Field, PrimeField};
 use rand::SeedableRng;
 use rand_xorshift::XorShiftRng;
+use std::ops::Neg;
 
 use crate::{CurveAffine, CurveProjective, EncodedPoint};
 
@@ -199,8 +200,7 @@ fn random_negation_tests<G: CurveProjective>() {
         let r = G::random(&mut rng);
 
         let s = G::Scalar::random(&mut rng);
-        let mut sneg = s;
-        sneg.negate();
+        let sneg = s.neg();
 
         let mut t1 = r;
         t1.mul_assign(s);
diff --git a/pairing/benches/bls12_381/fq.rs b/pairing/benches/bls12_381/fq.rs
index 3ed810aec..046491e28 100644
--- a/pairing/benches/bls12_381/fq.rs
+++ b/pairing/benches/bls12_381/fq.rs
@@ -1,6 +1,6 @@
 use rand_core::SeedableRng;
 use rand_xorshift::XorShiftRng;
-use std::ops::{AddAssign, MulAssign, SubAssign};
+use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
 use ff::{Field, PrimeField, PrimeFieldRepr, SqrtField};
 use pairing::bls12_381::*;
@@ -236,7 +236,7 @@ fn bench_fq_inverse(b: &mut ::test::Bencher) {
 }
 
 #[bench]
-fn bench_fq_negate(b: &mut ::test::Bencher) {
+fn bench_fq_neg(b: &mut ::test::Bencher) {
     const SAMPLES: usize = 1000;
 
     let mut rng = XorShiftRng::from_seed([
@@ -248,8 +248,7 @@ fn bench_fq_negate(b: &mut ::test::Bencher) {
 
     let mut count = 0;
     b.iter(|| {
-        let mut tmp = v[count];
-        tmp.negate();
+        let tmp = v[count].neg();
         count = (count + 1) % SAMPLES;
         tmp
     });
diff --git a/pairing/benches/bls12_381/fr.rs b/pairing/benches/bls12_381/fr.rs
index 590543194..16a5c0854 100644
--- a/pairing/benches/bls12_381/fr.rs
+++ b/pairing/benches/bls12_381/fr.rs
@@ -1,6 +1,6 @@
 use rand_core::SeedableRng;
 use rand_xorshift::XorShiftRng;
-use std::ops::{AddAssign, MulAssign, SubAssign};
+use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
 use ff::{Field, PrimeField, PrimeFieldRepr, SqrtField};
 use pairing::bls12_381::*;
@@ -236,7 +236,7 @@ fn bench_fr_inverse(b: &mut ::test::Bencher) {
 }
 
 #[bench]
-fn bench_fr_negate(b: &mut ::test::Bencher) {
+fn bench_fr_neg(b: &mut ::test::Bencher) {
     const SAMPLES: usize = 1000;
 
     let mut rng = XorShiftRng::from_seed([
@@ -248,8 +248,7 @@ fn bench_fr_negate(b: &mut ::test::Bencher) {
 
     let mut count = 0;
     b.iter(|| {
-        let mut tmp = v[count];
-        tmp.negate();
+        let tmp = v[count].neg();
         count = (count + 1) % SAMPLES;
         tmp
     });
diff --git a/pairing/src/bls12_381/ec.rs b/pairing/src/bls12_381/ec.rs
index 1d740b3a2..c1da107c5 100644
--- a/pairing/src/bls12_381/ec.rs
+++ b/pairing/src/bls12_381/ec.rs
@@ -107,8 +107,7 @@ macro_rules! curve_impl {
                 x3b.add_assign(&$affine::get_coeff_b());
 
                 x3b.sqrt().map(|y| {
-                    let mut negy = y;
-                    negy.negate();
+                    let negy = y.neg();
 
                     $affine {
                         x: x,
@@ -171,7 +170,7 @@ macro_rules! curve_impl {
 
             fn negate(&mut self) {
                 if !self.is_zero() {
-                    self.y.negate();
+                    self.y = self.y.neg();
                 }
             }
 
@@ -527,7 +526,7 @@ macro_rules! curve_impl {
 
             fn negate(&mut self) {
                 if !self.is_zero() {
-                    self.y.negate()
+                    self.y = self.y.neg();
                 }
             }
 
@@ -627,7 +626,7 @@ pub mod g1 {
     use group::{CurveAffine, CurveProjective, EncodedPoint, GroupDecodingError};
     use rand_core::RngCore;
     use std::fmt;
-    use std::ops::{AddAssign, MulAssign, SubAssign};
+    use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
     curve_impl!(
         "G1",
@@ -849,8 +848,7 @@ pub mod g1 {
                     affine.x.into_repr().write_be(&mut writer).unwrap();
                 }
 
-                let mut negy = affine.y;
-                negy.negate();
+                let negy = affine.y.neg();
 
                 // Set the third most significant bit if the correct y-coordinate
                 // is lexicographically largest.
@@ -948,8 +946,7 @@ pub mod g1 {
 
             if let Some(y) = rhs.sqrt() {
                 let yrepr = y.into_repr();
-                let mut negy = y;
-                negy.negate();
+                let negy = y.neg();
                 let negyrepr = negy.into_repr();
 
                 let p = G1Affine {
@@ -1297,7 +1294,7 @@ pub mod g2 {
     use group::{CurveAffine, CurveProjective, EncodedPoint, GroupDecodingError};
     use rand_core::RngCore;
     use std::fmt;
-    use std::ops::{AddAssign, MulAssign, SubAssign};
+    use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
     curve_impl!(
         "G2",
@@ -1544,8 +1541,7 @@ pub mod g2 {
                     affine.x.c0.into_repr().write_be(&mut writer).unwrap();
                 }
 
-                let mut negy = affine.y;
-                negy.negate();
+                let negy = affine.y.neg();
 
                 // Set the third most significant bit if the correct y-coordinate
                 // is lexicographically largest.
@@ -1654,8 +1650,7 @@ pub mod g2 {
             rhs.add_assign(&G2Affine::get_coeff_b());
 
             if let Some(y) = rhs.sqrt() {
-                let mut negy = y;
-                negy.negate();
+                let negy = y.neg();
 
                 let p = G2Affine {
                     x,
diff --git a/pairing/src/bls12_381/fq.rs b/pairing/src/bls12_381/fq.rs
index ce76b5434..f17a2783b 100644
--- a/pairing/src/bls12_381/fq.rs
+++ b/pairing/src/bls12_381/fq.rs
@@ -2,6 +2,9 @@ use super::fq2::Fq2;
 use ff::{Field, PrimeField, PrimeFieldDecodingError, PrimeFieldRepr};
 use std::ops::{AddAssign, MulAssign, SubAssign};
 
+#[cfg(test)]
+use std::ops::Neg;
+
 // B coefficient of BLS12-381 curve, 4.
 pub const B_COEFF: Fq = Fq(FqRepr([
     0xaa270000000cfff3,
@@ -456,8 +459,7 @@ fn test_b_coeff() {
 
 #[test]
 fn test_frob_coeffs() {
-    let mut nqr = Fq::one();
-    nqr.negate();
+    let nqr = Fq::one().neg();
 
     assert_eq!(FROBENIUS_COEFF_FQ2_C1[0], Fq::one());
     assert_eq!(
@@ -1167,8 +1169,7 @@ fn test_frob_coeffs() {
 
 #[test]
 fn test_neg_one() {
-    let mut o = Fq::one();
-    o.negate();
+    let o = Fq::one().neg();
 
     assert_eq!(NEGATIVE_ONE, o);
 }
@@ -2009,10 +2010,9 @@ fn test_fq_double() {
 }
 
 #[test]
-fn test_fq_negate() {
+fn test_fq_neg() {
     {
-        let mut a = Fq::zero();
-        a.negate();
+        let a = Fq::zero().neg();
 
         assert!(a.is_zero());
     }
@@ -2025,8 +2025,7 @@ fn test_fq_negate() {
     for _ in 0..1000 {
         // Ensure (a - (-a)) = 0.
         let mut a = Fq::random(&mut rng);
-        let mut b = a;
-        b.negate();
+        let b = a.neg();
         a.add_assign(&b);
 
         assert!(a.is_zero());
@@ -2074,8 +2073,7 @@ fn test_fq_sqrt() {
     for _ in 0..1000 {
         // Ensure sqrt(a^2) = a or -a
         let a = Fq::random(&mut rng);
-        let mut nega = a;
-        nega.negate();
+        let nega = a.neg();
         let mut b = a;
         b.square();
 
diff --git a/pairing/src/bls12_381/fq12.rs b/pairing/src/bls12_381/fq12.rs
index 96b452d74..6bc66a45e 100644
--- a/pairing/src/bls12_381/fq12.rs
+++ b/pairing/src/bls12_381/fq12.rs
@@ -3,7 +3,7 @@ use super::fq2::Fq2;
 use super::fq6::Fq6;
 use ff::Field;
 use rand_core::RngCore;
-use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};
+use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
 /// An element of Fq12, represented by c0 + c1 * w.
 #[derive(Copy, Clone, Debug, Eq, PartialEq)]
@@ -20,7 +20,7 @@ impl ::std::fmt::Display for Fq12 {
 
 impl Fq12 {
     pub fn conjugate(&mut self) {
-        self.c1.negate();
+        self.c1 = self.c1.neg();
     }
 
     pub fn mul_by_014(&mut self, c0: &Fq2, c1: &Fq2, c4: &Fq2) {
@@ -40,6 +40,17 @@ impl Fq12 {
     }
 }
 
+impl Neg for Fq12 {
+    type Output = Self;
+
+    fn neg(self) -> Self {
+        Fq12 {
+            c0: self.c0.neg(),
+            c1: self.c1.neg(),
+        }
+    }
+}
+
 impl<'r> Add<&'r Fq12> for Fq12 {
     type Output = Self;
 
@@ -177,11 +188,6 @@ impl Field for Fq12 {
         self.c1.double();
     }
 
-    fn negate(&mut self) {
-        self.c0.negate();
-        self.c1.negate();
-    }
-
     fn frobenius_map(&mut self, power: usize) {
         self.c0.frobenius_map(power);
         self.c1.frobenius_map(power);
@@ -216,13 +222,9 @@ impl Field for Fq12 {
         c1s.mul_by_nonresidue();
         c0s.sub_assign(&c1s);
 
-        c0s.inverse().map(|t| {
-            let mut tmp = Fq12 { c0: t, c1: t };
-            tmp.c0.mul_assign(&self.c0);
-            tmp.c1.mul_assign(&self.c1);
-            tmp.c1.negate();
-
-            tmp
+        c0s.inverse().map(|t| Fq12 {
+            c0: t.mul(&self.c0),
+            c1: t.mul(&self.c1).neg(),
         })
     }
 }
diff --git a/pairing/src/bls12_381/fq2.rs b/pairing/src/bls12_381/fq2.rs
index 699707269..e60c374d8 100644
--- a/pairing/src/bls12_381/fq2.rs
+++ b/pairing/src/bls12_381/fq2.rs
@@ -2,7 +2,7 @@ use super::fq::{Fq, FROBENIUS_COEFF_FQ2_C1, NEGATIVE_ONE};
 use ff::{Field, SqrtField};
 use rand_core::RngCore;
 use std::cmp::Ordering;
-use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};
+use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
 /// An element of Fq2, represented by c0 + c1 * u.
 #[derive(Copy, Clone, Debug, Eq, PartialEq)]
@@ -56,6 +56,17 @@ impl Fq2 {
     }
 }
 
+impl Neg for Fq2 {
+    type Output = Self;
+
+    fn neg(self) -> Self {
+        Fq2 {
+            c0: self.c0.neg(),
+            c1: self.c1.neg(),
+        }
+    }
+}
+
 impl<'r> Add<&'r Fq2> for Fq2 {
     type Output = Self;
 
@@ -192,8 +203,7 @@ impl Field for Fq2 {
         ab.mul_assign(&self.c1);
         let mut c0c1 = self.c0;
         c0c1.add_assign(&self.c1);
-        let mut c0 = self.c1;
-        c0.negate();
+        let mut c0 = self.c1.neg();
         c0.add_assign(&self.c0);
         c0.mul_assign(&c0c1);
         c0.sub_assign(&ab);
@@ -208,27 +218,15 @@ impl Field for Fq2 {
         self.c1.double();
     }
 
-    fn negate(&mut self) {
-        self.c0.negate();
-        self.c1.negate();
-    }
-
     fn inverse(&self) -> Option<Self> {
         let mut t1 = self.c1;
         t1.square();
         let mut t0 = self.c0;
         t0.square();
         t0.add_assign(&t1);
-        t0.inverse().map(|t| {
-            let mut tmp = Fq2 {
-                c0: self.c0,
-                c1: self.c1,
-            };
-            tmp.c0.mul_assign(&t);
-            tmp.c1.mul_assign(&t);
-            tmp.c1.negate();
-
-            tmp
+        t0.inverse().map(|t| Fq2 {
+            c0: self.c0.mul(&t),
+            c1: self.c1.mul(&t).neg(),
         })
     }
 
@@ -372,10 +370,8 @@ fn test_fq2_squaring() {
     }; // u
     a.square();
     assert_eq!(a, {
-        let mut neg1 = Fq::one();
-        neg1.negate();
         Fq2 {
-            c0: neg1,
+            c0: Fq::one().neg(),
             c1: Fq::zero(),
         }
     }); // -1
@@ -694,7 +690,7 @@ fn test_fq2_negation() {
     use super::fq::FqRepr;
     use ff::PrimeField;
 
-    let mut a = Fq2 {
+    let a = Fq2 {
         c0: Fq::from_repr(FqRepr([
             0x2d0078036923ffc7,
             0x11e59ea221a3b6d2,
@@ -713,8 +709,8 @@ fn test_fq2_negation() {
             0x12d1137b8a6a837,
         ]))
         .unwrap(),
-    };
-    a.negate();
+    }
+    .neg();
     assert_eq!(
         a,
         Fq2 {
@@ -1000,8 +996,7 @@ fn test_fq2_legendre() {
 
     assert_eq!(Zero, Fq2::zero().legendre());
     // i^2 = -1
-    let mut m1 = Fq2::one();
-    m1.negate();
+    let mut m1 = Fq2::one().neg();
     assert_eq!(QuadraticResidue, m1.legendre());
     m1.mul_by_nonresidue();
     assert_eq!(QuadraticNonResidue, m1.legendre());
diff --git a/pairing/src/bls12_381/fq6.rs b/pairing/src/bls12_381/fq6.rs
index 444119ac0..a50e283d3 100644
--- a/pairing/src/bls12_381/fq6.rs
+++ b/pairing/src/bls12_381/fq6.rs
@@ -2,7 +2,7 @@ use super::fq::{FROBENIUS_COEFF_FQ6_C1, FROBENIUS_COEFF_FQ6_C2};
 use super::fq2::Fq2;
 use ff::Field;
 use rand_core::RngCore;
-use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};
+use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
 /// An element of Fq6, represented by c0 + c1 * v + c2 * v^(2).
 #[derive(Copy, Clone, Debug, Eq, PartialEq)]
@@ -100,6 +100,18 @@ impl Fq6 {
     }
 }
 
+impl Neg for Fq6 {
+    type Output = Self;
+
+    fn neg(self) -> Self {
+        Fq6 {
+            c0: self.c0.neg(),
+            c1: self.c1.neg(),
+            c2: self.c2.neg(),
+        }
+    }
+}
+
 impl<'r> Add<&'r Fq6> for Fq6 {
     type Output = Self;
 
@@ -280,12 +292,6 @@ impl Field for Fq6 {
         self.c2.double();
     }
 
-    fn negate(&mut self) {
-        self.c0.negate();
-        self.c1.negate();
-        self.c2.negate();
-    }
-
     fn frobenius_map(&mut self, power: usize) {
         self.c0.frobenius_map(power);
         self.c1.frobenius_map(power);
@@ -332,7 +338,7 @@ impl Field for Fq6 {
         let mut c0 = self.c2;
         c0.mul_by_nonresidue();
         c0.mul_assign(&self.c1);
-        c0.negate();
+        c0 = c0.neg();
         {
             let mut c0s = self.c0;
             c0s.square();
diff --git a/pairing/src/bls12_381/fr.rs b/pairing/src/bls12_381/fr.rs
index ca9cd575a..d72557e6f 100644
--- a/pairing/src/bls12_381/fr.rs
+++ b/pairing/src/bls12_381/fr.rs
@@ -10,6 +10,8 @@ pub struct Fr(FrRepr);
 use rand_core::SeedableRng;
 #[cfg(test)]
 use rand_xorshift::XorShiftRng;
+#[cfg(test)]
+use std::ops::Neg;
 
 #[test]
 fn test_fr_repr_ordering() {
@@ -767,10 +769,9 @@ fn test_fr_double() {
 }
 
 #[test]
-fn test_fr_negate() {
+fn test_fr_neg() {
     {
-        let mut a = Fr::zero();
-        a.negate();
+        let a = Fr::zero().neg();
 
         assert!(a.is_zero());
     }
@@ -783,8 +784,7 @@ fn test_fr_negate() {
     for _ in 0..1000 {
         // Ensure (a - (-a)) = 0.
         let mut a = Fr::random(&mut rng);
-        let mut b = a;
-        b.negate();
+        let b = a.neg();
         a.add_assign(&b);
 
         assert!(a.is_zero());
@@ -832,8 +832,7 @@ fn test_fr_sqrt() {
     for _ in 0..1000 {
         // Ensure sqrt(a^2) = a or -a
         let a = Fr::random(&mut rng);
-        let mut nega = a;
-        nega.negate();
+        let nega = a.neg();
         let mut b = a;
         b.square();
 
diff --git a/pairing/src/bls12_381/mod.rs b/pairing/src/bls12_381/mod.rs
index 04fd5ee5d..32e23e1a5 100644
--- a/pairing/src/bls12_381/mod.rs
+++ b/pairing/src/bls12_381/mod.rs
@@ -25,7 +25,7 @@ use super::{Engine, PairingCurveAffine};
 
 use ff::{BitIterator, Field, ScalarEngine};
 use group::CurveAffine;
-use std::ops::{AddAssign, MulAssign, SubAssign};
+use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
 // The BLS parameter x for BLS12-381 is -0xd201000000010000
 const BLS_X: u64 = 0xd201000000010000;
@@ -236,7 +236,7 @@ impl G2Prepared {
             tmp3 = tmp4;
             tmp3.mul_assign(&zsquared);
             tmp3.double();
-            tmp3.negate();
+            tmp3 = tmp3.neg();
 
             tmp6.square();
             tmp6.sub_assign(&tmp0);
@@ -334,7 +334,7 @@ impl G2Prepared {
             t10 = r.z;
             t10.double();
 
-            t6.negate();
+            t6 = t6.neg();
 
             t1 = t6;
             t1.double();
diff --git a/pairing/src/tests/field.rs b/pairing/src/tests/field.rs
index 8f3d8d9e0..89b203887 100644
--- a/pairing/src/tests/field.rs
+++ b/pairing/src/tests/field.rs
@@ -36,8 +36,7 @@ pub fn random_sqrt_tests<F: SqrtField>() {
         assert_eq!(b.legendre(), LegendreSymbol::QuadraticResidue);
 
         let b = b.sqrt().unwrap();
-        let mut negb = b;
-        negb.negate();
+        let negb = b.neg();
 
         assert!(a == b || a == negb);
     }
@@ -51,7 +50,7 @@ pub fn random_sqrt_tests<F: SqrtField>() {
         b = b.sqrt().unwrap();
 
         if b != c {
-            b.negate();
+            b = b.neg();
         }
 
         assert_eq!(b, c);
@@ -77,8 +76,7 @@ pub fn random_field_tests<F: Field>() {
 
     assert!(F::zero().is_zero());
     {
-        let mut z = F::zero();
-        z.negate();
+        let z = F::zero().neg();
         assert!(z.is_zero());
     }
 
@@ -204,8 +202,7 @@ fn random_subtraction_tests<F: Field, R: RngCore>(rng: &mut R) {
 fn random_negation_tests<F: Field, R: RngCore>(rng: &mut R) {
     for _ in 0..10000 {
         let a = F::random(rng);
-        let mut b = a;
-        b.negate();
+        let mut b = a.neg();
         b.add_assign(&a);
 
         assert!(b.is_zero());
diff --git a/zcash_primitives/src/jubjub/edwards.rs b/zcash_primitives/src/jubjub/edwards.rs
index 2865ce3d4..8c7df4c71 100644
--- a/zcash_primitives/src/jubjub/edwards.rs
+++ b/zcash_primitives/src/jubjub/edwards.rs
@@ -1,5 +1,5 @@
 use ff::{BitIterator, Field, PrimeField, PrimeFieldRepr, SqrtField};
-use std::ops::{AddAssign, MulAssign, SubAssign};
+use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
 use super::{montgomery, JubjubEngine, JubjubParams, PrimeOrder, Unknown};
 
@@ -126,7 +126,7 @@ impl<E: JubjubEngine> Point<E, Unknown> {
                 match tmp1.sqrt() {
                     Some(mut x) => {
                         if x.into_repr().is_odd() != sign {
-                            x.negate();
+                            x = x.neg();
                         }
 
                         let mut t = x;
@@ -213,12 +213,9 @@ impl<E: JubjubEngine, Subgroup> Point<E, Subgroup> {
                 // only point of order 2 that is not the neutral element.
                 if y.is_zero() {
                     // This must be the point (0, 0) as above.
-                    let mut neg1 = E::Fr::one();
-                    neg1.negate();
-
                     Point {
                         x: E::Fr::zero(),
-                        y: neg1,
+                        y: E::Fr::one().neg(),
                         t: E::Fr::zero(),
                         z: E::Fr::one(),
                         _marker: PhantomData,
@@ -324,8 +321,8 @@ impl<E: JubjubEngine, Subgroup> Point<E, Subgroup> {
     pub fn negate(&self) -> Self {
         let mut p = self.clone();
 
-        p.x.negate();
-        p.t.negate();
+        p.x = p.x.neg();
+        p.t = p.t.neg();
 
         p
     }
@@ -352,8 +349,7 @@ impl<E: JubjubEngine, Subgroup> Point<E, Subgroup> {
 
         // D = a*A
         //   = -A
-        let mut d = a;
-        d.negate();
+        let d = a.neg();
 
         // E = (X1+Y1)^2 - A - B
         let mut e = self.x;
diff --git a/zcash_primitives/src/jubjub/fs.rs b/zcash_primitives/src/jubjub/fs.rs
index ddd2bceb1..53d0ee355 100644
--- a/zcash_primitives/src/jubjub/fs.rs
+++ b/zcash_primitives/src/jubjub/fs.rs
@@ -5,7 +5,7 @@ use ff::{
     PrimeField, PrimeFieldDecodingError, PrimeFieldRepr, SqrtField,
 };
 use rand_core::RngCore;
-use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};
+use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
 use super::ToUniform;
 
@@ -269,6 +269,20 @@ impl From<Fs> for FsRepr {
     }
 }
 
+impl Neg for Fs {
+    type Output = Self;
+
+    #[inline]
+    fn neg(mut self) -> Self {
+        if !self.is_zero() {
+            let mut tmp = MODULUS;
+            tmp.sub_noborrow(&self.0);
+            self.0 = tmp;
+        }
+        self
+    }
+}
+
 impl<'r> Add<&'r Fs> for Fs {
     type Output = Self;
 
@@ -496,15 +510,6 @@ impl Field for Fs {
         self.reduce();
     }
 
-    #[inline]
-    fn negate(&mut self) {
-        if !self.is_zero() {
-            let mut tmp = MODULUS;
-            tmp.sub_noborrow(&self.0);
-            self.0 = tmp;
-        }
-    }
-
     fn inverse(&self) -> Option<Self> {
         if self.is_zero() {
             None
@@ -742,8 +747,7 @@ impl SqrtField for Fs {
 
 #[test]
 fn test_neg_one() {
-    let mut o = Fs::one();
-    o.negate();
+    let o = Fs::one().neg();
 
     assert_eq!(NEGATIVE_ONE, o);
 }
@@ -1471,10 +1475,9 @@ fn test_fs_double() {
 }
 
 #[test]
-fn test_fs_negate() {
+fn test_fs_neg() {
     {
-        let mut a = Fs::zero();
-        a.negate();
+        let a = Fs::zero().neg();
 
         assert!(a.is_zero());
     }
@@ -1487,8 +1490,7 @@ fn test_fs_negate() {
     for _ in 0..1000 {
         // Ensure (a - (-a)) = 0.
         let mut a = Fs::random(&mut rng);
-        let mut b = a;
-        b.negate();
+        let b = a.neg();
         a.add_assign(&b);
 
         assert!(a.is_zero());
@@ -1534,8 +1536,7 @@ fn test_fs_sqrt() {
     for _ in 0..1000 {
         // Ensure sqrt(a^2) = a or -a
         let a = Fs::random(&mut rng);
-        let mut nega = a;
-        nega.negate();
+        let nega = a.neg();
         let mut b = a;
         b.square();
 
diff --git a/zcash_primitives/src/jubjub/montgomery.rs b/zcash_primitives/src/jubjub/montgomery.rs
index fc412742d..18acc0d50 100644
--- a/zcash_primitives/src/jubjub/montgomery.rs
+++ b/zcash_primitives/src/jubjub/montgomery.rs
@@ -1,5 +1,5 @@
 use ff::{BitIterator, Field, PrimeField, PrimeFieldRepr, SqrtField};
-use std::ops::{AddAssign, MulAssign, SubAssign};
+use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
 use super::{edwards, JubjubEngine, JubjubParams, PrimeOrder, Unknown};
 
@@ -62,7 +62,7 @@ impl<E: JubjubEngine> Point<E, Unknown> {
         match rhs.sqrt() {
             Some(mut y) => {
                 if y.into_repr().is_odd() != sign {
-                    y.negate();
+                    y = y.neg();
                 }
 
                 Some(Point {
@@ -190,7 +190,7 @@ impl<E: JubjubEngine, Subgroup> Point<E, Subgroup> {
     pub fn negate(&self) -> Self {
         let mut p = self.clone();
 
-        p.y.negate();
+        p.y = p.y.neg();
 
         p
     }
@@ -242,7 +242,7 @@ impl<E: JubjubEngine, Subgroup> Point<E, Subgroup> {
         y3.sub_assign(&self.x);
         y3.mul_assign(&delta);
         y3.add_assign(&self.y);
-        y3.negate();
+        y3 = y3.neg();
 
         Point {
             x: x3,
@@ -292,7 +292,7 @@ impl<E: JubjubEngine, Subgroup> Point<E, Subgroup> {
                     y3.sub_assign(&self.x);
                     y3.mul_assign(&delta);
                     y3.add_assign(&self.y);
-                    y3.negate();
+                    y3 = y3.neg();
 
                     Point {
                         x: x3,
diff --git a/zcash_primitives/src/jubjub/tests.rs b/zcash_primitives/src/jubjub/tests.rs
index 56fad56ab..d1527378f 100644
--- a/zcash_primitives/src/jubjub/tests.rs
+++ b/zcash_primitives/src/jubjub/tests.rs
@@ -1,7 +1,7 @@
 use super::{edwards, montgomery, JubjubEngine, JubjubParams, PrimeOrder};
 
 use ff::{Field, LegendreSymbol, PrimeField, PrimeFieldRepr, SqrtField};
-use std::ops::{AddAssign, MulAssign, SubAssign};
+use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
 use rand_core::{RngCore, SeedableRng};
 use rand_xorshift::XorShiftRng;
@@ -310,8 +310,7 @@ fn test_back_and_forth<E: JubjubEngine>(params: &E::Params) {
 
 fn test_jubjub_params<E: JubjubEngine>(params: &E::Params) {
     // a = -1
-    let mut a = E::Fr::one();
-    a.negate();
+    let a = E::Fr::one().neg();
 
     {
         // Check that 2A is consistent with A
@@ -339,7 +338,7 @@ fn test_jubjub_params<E: JubjubEngine>(params: &E::Params) {
         assert!(tmp.inverse().unwrap().legendre() == LegendreSymbol::QuadraticNonResidue);
 
         // tmp = -d
-        tmp.negate();
+        tmp = tmp.neg();
 
         // -d is nonsquare
         assert!(tmp.legendre() == LegendreSymbol::QuadraticNonResidue);
diff --git a/zcash_primitives/src/pedersen_hash.rs b/zcash_primitives/src/pedersen_hash.rs
index 5fb73339f..d12b6a407 100644
--- a/zcash_primitives/src/pedersen_hash.rs
+++ b/zcash_primitives/src/pedersen_hash.rs
@@ -2,7 +2,7 @@
 
 use crate::jubjub::*;
 use ff::{Field, PrimeField, PrimeFieldRepr};
-use std::ops::AddAssign;
+use std::ops::{AddAssign, Neg};
 
 #[derive(Copy, Clone)]
 pub enum Personalization {
@@ -65,7 +65,7 @@ where
 
             // conditionally negate
             if c {
-                tmp.negate();
+                tmp = tmp.neg();
             }
 
             acc.add_assign(&tmp);
diff --git a/zcash_primitives/src/redjubjub.rs b/zcash_primitives/src/redjubjub.rs
index 61586856e..4633d6d6f 100644
--- a/zcash_primitives/src/redjubjub.rs
+++ b/zcash_primitives/src/redjubjub.rs
@@ -7,7 +7,7 @@ use crate::jubjub::{edwards::Point, FixedGenerators, JubjubEngine, JubjubParams,
 use ff::{Field, PrimeField, PrimeFieldRepr};
 use rand_core::RngCore;
 use std::io::{self, Read, Write};
-use std::ops::{AddAssign, MulAssign};
+use std::ops::{AddAssign, MulAssign, Neg};
 
 use crate::util::hash_to_scalar;
 
@@ -194,7 +194,7 @@ pub fn batch_verify<'a, E: JubjubEngine, R: RngCore>(
         let z = E::Fs::random(rng);
 
         s.mul_assign(&z);
-        s.negate();
+        s = s.neg();
 
         r = r.mul(z, params);
 
diff --git a/zcash_proofs/src/circuit/ecc.rs b/zcash_proofs/src/circuit/ecc.rs
index a35dfd779..7072f0147 100644
--- a/zcash_proofs/src/circuit/ecc.rs
+++ b/zcash_proofs/src/circuit/ecc.rs
@@ -2,7 +2,7 @@
 
 use ff::Field;
 use pairing::Engine;
-use std::ops::{AddAssign, MulAssign, SubAssign};
+use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
 use bellman::{ConstraintSystem, SynthesisError};
 
@@ -367,7 +367,7 @@ impl<E: JubjubEngine> EdwardsPoint<E> {
         let y3 = AllocatedNum::alloc(cs.namespace(|| "y3"), || {
             let mut t0 = *a.get_value().get()?;
             t0.double();
-            t0.negate();
+            t0 = t0.neg();
             t0.add_assign(t.get_value().get()?);
 
             let mut t1 = E::Fr::one();
@@ -642,7 +642,7 @@ impl<E: JubjubEngine> MontgomeryPoint<E> {
             t0.sub_assign(self.x.get_value().get()?);
             t0.mul_assign(lambda.get_value().get()?);
             t0.add_assign(self.y.get_value().get()?);
-            t0.negate();
+            t0 = t0.neg();
 
             Ok(t0)
         })?;
diff --git a/zcash_proofs/src/sapling/prover.rs b/zcash_proofs/src/sapling/prover.rs
index e695ba12f..16424cca8 100644
--- a/zcash_proofs/src/sapling/prover.rs
+++ b/zcash_proofs/src/sapling/prover.rs
@@ -5,7 +5,7 @@ use bellman::{
 use ff::Field;
 use pairing::bls12_381::{Bls12, Fr};
 use rand_core::OsRng;
-use std::ops::AddAssign;
+use std::ops::{AddAssign, Neg};
 use zcash_primitives::{
     jubjub::{edwards, fs::Fs, FixedGenerators, JubjubBls12, Unknown},
     primitives::{Diversifier, Note, PaymentAddress, ProofGenerationKey, ValueCommitment},
@@ -202,8 +202,7 @@ impl SaplingProvingContext {
 
         // Accumulate the value commitment randomness in the context
         {
-            let mut tmp = rcv;
-            tmp.negate(); // Outputs subtract from the total.
+            let mut tmp = rcv.neg(); // Outputs subtract from the total.
             tmp.add_assign(&self.bsk);
 
             // Update the context