diff --git a/algebra/Cargo.toml b/algebra/Cargo.toml
index e3864bb38..5a260186b 100644
--- a/algebra/Cargo.toml
+++ b/algebra/Cargo.toml
@@ -25,6 +25,7 @@ edition = "2018"
 byteorder = { version = "1" }
 rand = { version = "0.7" }
 derivative = { version = "1" }
+num-traits = { version = "0.2.11"}
 
 colored = { version = "1", optional = true }
 rayon = { version = "1", optional = true }
diff --git a/algebra/src/curves/bls12_377/g1.rs b/algebra/src/curves/bls12_377/g1.rs
index 3805e6c8e..6b97f0838 100644
--- a/algebra/src/curves/bls12_377/g1.rs
+++ b/algebra/src/curves/bls12_377/g1.rs
@@ -4,9 +4,9 @@ use crate::{
     curves::models::{ModelParameters, SWModelParameters},
     fields::{
         bls12_377::{Fq, Fr},
-        Field,
     },
 };
+use num_traits::Zero;
 
 #[derive(Copy, Clone, Default, PartialEq, Eq)]
 pub struct Bls12_377G1Parameters;
diff --git a/algebra/src/curves/bls12_377/g2.rs b/algebra/src/curves/bls12_377/g2.rs
index c1a613772..121782f58 100644
--- a/algebra/src/curves/bls12_377/g2.rs
+++ b/algebra/src/curves/bls12_377/g2.rs
@@ -5,9 +5,9 @@ use crate::{
     curves::models::{ModelParameters, SWModelParameters},
     fields::{
         bls12_377::{Fq, Fq2, Fr},
-        Field,
     },
 };
+use num_traits::Zero;
 
 #[derive(Copy, Clone, Default, PartialEq, Eq)]
 pub struct Bls12_377G2Parameters;
diff --git a/algebra/src/curves/bls12_377/tests.rs b/algebra/src/curves/bls12_377/tests.rs
index b256018d9..653547a8e 100644
--- a/algebra/src/curves/bls12_377/tests.rs
+++ b/algebra/src/curves/bls12_377/tests.rs
@@ -14,6 +14,7 @@ use crate::{
     },
     groups::tests::group_test,
 };
+use num_traits::{One, Zero};
 use std::ops::{AddAssign, MulAssign};
 
 #[test]
diff --git a/algebra/src/curves/bls12_381/g1.rs b/algebra/src/curves/bls12_381/g1.rs
index 0271f32fc..65f04c224 100644
--- a/algebra/src/curves/bls12_381/g1.rs
+++ b/algebra/src/curves/bls12_381/g1.rs
@@ -9,9 +9,9 @@ use crate::{
     },
     fields::{
         bls12_381::{Fq, Fq12, Fr},
-        Field,
     },
 };
+use num_traits::Zero;
 
 pub type G1Affine = Bls12G1Affine<Bls12_381Parameters>;
 pub type G1Projective = Bls12G1Projective<Bls12_381Parameters>;
diff --git a/algebra/src/curves/bls12_381/g2.rs b/algebra/src/curves/bls12_381/g2.rs
index 112e04096..ed01eb9d7 100644
--- a/algebra/src/curves/bls12_381/g2.rs
+++ b/algebra/src/curves/bls12_381/g2.rs
@@ -12,9 +12,9 @@ use crate::{
     },
     fields::{
         bls12_381::{Fq, Fq12, Fq2, Fr},
-        Field,
     },
 };
+use num_traits::Zero;
 
 pub type G2Affine = Bls12G2Affine<Bls12_381Parameters>;
 pub type G2Projective = Bls12G2Projective<Bls12_381Parameters>;
diff --git a/algebra/src/curves/bls12_381/tests.rs b/algebra/src/curves/bls12_381/tests.rs
index 5c23947ae..910902c07 100644
--- a/algebra/src/curves/bls12_381/tests.rs
+++ b/algebra/src/curves/bls12_381/tests.rs
@@ -16,6 +16,7 @@ use crate::{
     },
     groups::tests::group_test,
 };
+use num_traits::{One, Zero};
 use rand;
 use std::ops::{AddAssign, MulAssign};
 
diff --git a/algebra/src/curves/jubjub/tests.rs b/algebra/src/curves/jubjub/tests.rs
index b3c58ef61..c038091dc 100644
--- a/algebra/src/curves/jubjub/tests.rs
+++ b/algebra/src/curves/jubjub/tests.rs
@@ -1,9 +1,13 @@
 use crate::{
     bytes::{FromBytes, ToBytes},
-    curves::{jubjub::*, tests::curve_tests, AffineCurve, ProjectiveCurve, models::twisted_edwards_extended::tests::montgomery_conversion_test},
+    curves::{
+        jubjub::*, models::twisted_edwards_extended::tests::montgomery_conversion_test,
+        tests::curve_tests, AffineCurve, ProjectiveCurve,
+    },
     fields::jubjub::fr::Fr,
     groups::tests::group_test,
 };
+use num_traits::Zero;
 use rand;
 use std::str::FromStr;
 
diff --git a/algebra/src/curves/mnt6/mod.rs b/algebra/src/curves/mnt6/mod.rs
index 08b05c199..e3396c25f 100644
--- a/algebra/src/curves/mnt6/mod.rs
+++ b/algebra/src/curves/mnt6/mod.rs
@@ -10,6 +10,7 @@ use crate::{
         BitIterator, Field, FpParameters,
     },
 };
+use num_traits::{One, Zero};
 
 pub mod g1;
 pub mod g2;
diff --git a/algebra/src/curves/mnt6/tests.rs b/algebra/src/curves/mnt6/tests.rs
index e702daf48..88c6e4f14 100644
--- a/algebra/src/curves/mnt6/tests.rs
+++ b/algebra/src/curves/mnt6/tests.rs
@@ -7,6 +7,7 @@ use crate::{
     fields::mnt6::fr::Fr,
     groups::tests::group_test,
 };
+use num_traits::One;
 use rand;
 
 #[test]
diff --git a/algebra/src/curves/mod.rs b/algebra/src/curves/mod.rs
index 3d295fb11..4f4f715df 100644
--- a/algebra/src/curves/mod.rs
+++ b/algebra/src/curves/mod.rs
@@ -4,6 +4,7 @@ use crate::{
     groups::Group,
 };
 use crate::UniformRand;
+use num_traits::Zero;
 use std::{
     fmt::{Debug, Display},
     hash::Hash,
@@ -126,9 +127,11 @@ pub trait ProjectiveCurve:
     + Debug
     + Display
     + UniformRand
+    + Zero
     + 'static
     + Neg<Output = Self>
     + for<'a> Add<&'a Self, Output = Self>
+    + Add<Self, Output = Self>
     + for<'a> Sub<&'a Self, Output = Self>
     + for<'a> AddAssign<&'a Self>
     + for<'a> SubAssign<&'a Self>
@@ -137,18 +140,10 @@ pub trait ProjectiveCurve:
     type BaseField: Field;
     type Affine: AffineCurve<Projective = Self, ScalarField = Self::ScalarField>;
 
-    /// Returns the additive identity.
-    #[must_use]
-    fn zero() -> Self;
-
     /// Returns a fixed generator of unknown exponent.
     #[must_use]
     fn prime_subgroup_generator() -> Self;
 
-    /// Determines if this point is the point at infinity.
-    #[must_use]
-    fn is_zero(&self) -> bool;
-
     /// Normalizes a slice of projective elements so that
     /// conversion to affine is cheap.
     fn batch_normalization(v: &mut [Self]);
@@ -205,6 +200,7 @@ pub trait AffineCurve:
     + Hash
     + Debug
     + Display
+    + Zero
     + Neg<Output = Self>
     + 'static
 {
@@ -212,19 +208,10 @@ pub trait AffineCurve:
     type BaseField: Field;
     type Projective: ProjectiveCurve<Affine = Self, ScalarField = Self::ScalarField>;
 
-    /// Returns the additive identity.
-    #[must_use]
-    fn zero() -> Self;
-
     /// Returns a fixed generator of unknown exponent.
     #[must_use]
     fn prime_subgroup_generator() -> Self;
 
-    /// Determines if this point represents the point at infinity; the
-    /// additive identity.
-    #[must_use]
-    fn is_zero(&self) -> bool;
-
     /// Performs scalar multiplication of this element with mixed addition.
     #[must_use]
     fn mul<S: Into<<Self::ScalarField as PrimeField>::BigInt>>(&self, other: S)
@@ -261,15 +248,6 @@ pub trait PairingCurve: AffineCurve {
 
 impl<C: ProjectiveCurve> Group for C {
     type ScalarField = C::ScalarField;
-    #[must_use]
-    fn zero() -> Self {
-        <C as ProjectiveCurve>::zero()
-    }
-
-    #[must_use]
-    fn is_zero(&self) -> bool {
-        <C as ProjectiveCurve>::is_zero(&self)
-    }
 
     #[inline]
     #[must_use]
diff --git a/algebra/src/curves/models/bls12/g1.rs b/algebra/src/curves/models/bls12/g1.rs
index d8bed2456..4f0494749 100644
--- a/algebra/src/curves/models/bls12/g1.rs
+++ b/algebra/src/curves/models/bls12/g1.rs
@@ -6,6 +6,7 @@ use crate::{
         AffineCurve,
     },
 };
+use num_traits::Zero;
 use std::io::{Result as IoResult, Write};
 
 pub type G1Affine<P> = GroupAffine<<P as Bls12Parameters>::G1Parameters>;
diff --git a/algebra/src/curves/models/bls12/g2.rs b/algebra/src/curves/models/bls12/g2.rs
index 3c4235f0c..03cd9a3e1 100644
--- a/algebra/src/curves/models/bls12/g2.rs
+++ b/algebra/src/curves/models/bls12/g2.rs
@@ -8,6 +8,7 @@ use crate::{
     },
     fields::{BitIterator, Field, Fp2},
 };
+use num_traits::{One, Zero};
 use std::io::{Result as IoResult, Write};
 
 pub type G2Affine<P> = GroupAffine<<P as Bls12Parameters>::G2Parameters>;
diff --git a/algebra/src/curves/models/bls12/mod.rs b/algebra/src/curves/models/bls12/mod.rs
index bc3c22772..3fe24475d 100644
--- a/algebra/src/curves/models/bls12/mod.rs
+++ b/algebra/src/curves/models/bls12/mod.rs
@@ -10,6 +10,7 @@ use crate::{
         BitIterator, Field, Fp2, PrimeField, SquareRootField,
     },
 };
+use num_traits::One;
 
 use std::marker::PhantomData;
 
diff --git a/algebra/src/curves/models/short_weierstrass_jacobian.rs b/algebra/src/curves/models/short_weierstrass_jacobian.rs
index 728eb3cba..d9f9d379e 100644
--- a/algebra/src/curves/models/short_weierstrass_jacobian.rs
+++ b/algebra/src/curves/models/short_weierstrass_jacobian.rs
@@ -1,6 +1,7 @@
 use crate::curves::models::SWModelParameters as Parameters;
 use rand::{Rng, distributions::{Standard, Distribution}};
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use std::{
     fmt::{Display, Formatter, Result as FmtResult},
     io::{Read, Result as IoResult, Write},
@@ -102,16 +103,43 @@ impl<P: Parameters> GroupAffine<P> {
     }
 }
 
+impl<P: Parameters> Zero for GroupAffine<P> {
+    #[inline]
+    fn zero() -> Self {
+        Self::new(P::BaseField::zero(), P::BaseField::one(), true)
+    }
+
+    #[inline]
+    fn is_zero(&self) -> bool {
+        self.infinity
+    }
+}
+
+impl<P: Parameters> Add<Self> for GroupAffine<P> {
+    type Output = Self;
+    fn add(self, other: Self) -> Self {
+        let mut copy = self;
+        copy += &other;
+        copy
+    }
+}
+
+impl<'a, P: Parameters> AddAssign<&'a Self> for GroupAffine<P> {
+    fn add_assign(&mut self, other: &'a Self) {
+        let lambda = (other.y - &self.y) / &(other.x - &self.y);
+        let x3 = lambda * lambda - &self.x - &other.x;
+        let y3 = (self.x - &x3) * lambda - &self.y;
+
+        self.x = x3;
+        self.y = y3;
+    }
+}
+
 impl<P: Parameters> AffineCurve for GroupAffine<P> {
     type BaseField = P::BaseField;
     type ScalarField = P::ScalarField;
     type Projective = GroupProjective<P>;
 
-    #[inline]
-    fn zero() -> Self {
-        Self::new(Self::BaseField::zero(), Self::BaseField::one(), true)
-    }
-
     #[inline]
     fn prime_subgroup_generator() -> Self {
         Self::new(
@@ -121,11 +149,6 @@ impl<P: Parameters> AffineCurve for GroupAffine<P> {
         )
     }
 
-    #[inline]
-    fn is_zero(&self) -> bool {
-        self.infinity
-    }
-
     #[inline]
     fn mul<S: Into<<Self::ScalarField as PrimeField>::BigInt>>(&self, by: S) -> GroupProjective<P> {
         let bits = BitIterator::new(by.into());
@@ -238,8 +261,6 @@ impl<P: Parameters> Distribution<GroupProjective<P>> for Standard {
     }
 }
 
-
-
 impl<P: Parameters> ToBytes for GroupProjective<P> {
     #[inline]
     fn write<W: Write>(&self, mut writer: W) -> IoResult<()> {
@@ -277,11 +298,7 @@ impl<P: Parameters> GroupProjective<P> {
     }
 }
 
-impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
-    type BaseField = P::BaseField;
-    type ScalarField = P::ScalarField;
-    type Affine = GroupAffine<P>;
-
+impl<P: Parameters> Zero for GroupProjective<P> {
     // The point at infinity is always represented by
     // Z = 0.
     #[inline]
@@ -293,17 +310,23 @@ impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
         )
     }
 
-    #[inline]
-    fn prime_subgroup_generator() -> Self {
-        GroupAffine::prime_subgroup_generator().into()
-    }
-
     // The point at infinity is always represented by
     // Z = 0.
     #[inline]
     fn is_zero(&self) -> bool {
         self.z.is_zero()
     }
+}
+
+impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
+    type BaseField = P::BaseField;
+    type ScalarField = P::ScalarField;
+    type Affine = GroupAffine<P>;
+
+    #[inline]
+    fn prime_subgroup_generator() -> Self {
+        GroupAffine::prime_subgroup_generator().into()
+    }
 
     #[inline]
     fn is_normalized(&self) -> bool {
@@ -557,13 +580,13 @@ impl<P: Parameters> Neg for GroupProjective<P> {
     }
 }
 
-impl<'a, P: Parameters> Add<&'a Self> for GroupProjective<P> {
+impl<P: Parameters> Add<Self> for GroupProjective<P> {
     type Output = Self;
 
     #[inline]
-    fn add(self, other: &'a Self) -> Self {
+    fn add(self, other: Self) -> Self {
         let mut copy = self;
-        copy += other;
+        copy += &other;
         copy
     }
 }
diff --git a/algebra/src/curves/models/short_weierstrass_projective.rs b/algebra/src/curves/models/short_weierstrass_projective.rs
index 97ac508af..18f749894 100644
--- a/algebra/src/curves/models/short_weierstrass_projective.rs
+++ b/algebra/src/curves/models/short_weierstrass_projective.rs
@@ -1,6 +1,7 @@
 use crate::curves::models::SWModelParameters as Parameters;
 use rand::{Rng, distributions::{Standard, Distribution}};
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use std::{
     fmt::{Display, Formatter, Result as FmtResult},
     io::{Read, Result as IoResult, Write},
@@ -107,15 +108,41 @@ impl<P: Parameters> GroupAffine<P> {
     }
 }
 
+impl<P: Parameters> Zero for GroupAffine<P> {
+    fn zero() -> Self {
+        Self::new(P::BaseField::zero(), P::BaseField::one(), true)
+    }
+
+    fn is_zero(&self) -> bool {
+        self.infinity
+    }
+}
+
+impl<P: Parameters> Add<Self> for GroupAffine<P> {
+    type Output = Self;
+    fn add(self, other: Self) -> Self {
+        let mut copy = self;
+        copy += &other;
+        copy
+    }
+}
+
+impl<'a, P: Parameters> AddAssign<&'a Self> for GroupAffine<P> {
+    fn add_assign(&mut self, other: &'a Self) {
+        let lambda = (other.y - &self.y) / &(other.x - &self.y);
+        let x3 = lambda * lambda - &self.x - &other.x;
+        let y3 = (self.x - &x3) * lambda - &self.y;
+
+        self.x = x3;
+        self.y = y3;
+    }
+}
+
 impl<P: Parameters> AffineCurve for GroupAffine<P> {
     type BaseField = P::BaseField;
     type ScalarField = P::ScalarField;
     type Projective = GroupProjective<P>;
 
-    fn zero() -> Self {
-        Self::new(Self::BaseField::zero(), Self::BaseField::one(), true)
-    }
-
     fn prime_subgroup_generator() -> Self {
         Self::new(
             P::AFFINE_GENERATOR_COEFFS.0,
@@ -124,10 +151,6 @@ impl<P: Parameters> AffineCurve for GroupAffine<P> {
         )
     }
 
-    fn is_zero(&self) -> bool {
-        self.infinity
-    }
-
     fn mul<S: Into<<Self::ScalarField as PrimeField>::BigInt>>(&self, by: S) -> GroupProjective<P> {
         let bits = BitIterator::new(by.into());
         self.mul_bits(bits)
@@ -231,9 +254,6 @@ impl<P: Parameters> Distribution<GroupProjective<P>> for Standard {
     }
 }
 
-
-
-
 impl<P: Parameters> ToBytes for GroupProjective<P> {
     #[inline]
     fn write<W: Write>(&self, mut writer: W) -> IoResult<()> {
@@ -271,11 +291,7 @@ impl<P: Parameters> GroupProjective<P> {
     }
 }
 
-impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
-    type BaseField = P::BaseField;
-    type ScalarField = P::ScalarField;
-    type Affine = GroupAffine<P>;
-
+impl<P: Parameters> Zero for GroupProjective<P> {
     // The point at infinity is always represented by Z = 0.
     #[inline]
     fn zero() -> Self {
@@ -286,17 +302,23 @@ impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
         )
     }
 
-    #[inline]
-    fn prime_subgroup_generator() -> Self {
-        GroupAffine::prime_subgroup_generator().into()
-    }
-
     // The point at infinity is always represented by
     // Z = 0.
     #[inline]
     fn is_zero(&self) -> bool {
         self.z.is_zero()
     }
+}
+
+impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
+    type BaseField = P::BaseField;
+    type ScalarField = P::ScalarField;
+    type Affine = GroupAffine<P>;
+
+    #[inline]
+    fn prime_subgroup_generator() -> Self {
+        GroupAffine::prime_subgroup_generator().into()
+    }
 
     #[inline]
     fn is_normalized(&self) -> bool {
@@ -466,11 +488,11 @@ impl<P: Parameters> Neg for GroupProjective<P> {
     }
 }
 
-impl<'a, P: Parameters> Add<&'a Self> for GroupProjective<P> {
+impl<P: Parameters> Add<Self> for GroupProjective<P> {
     type Output = Self;
-    fn add(self, other: &'a Self) -> Self {
+    fn add(self, other: Self) -> Self {
         let mut copy = self;
-        copy += other;
+        copy += &other;
         copy
     }
 }
diff --git a/algebra/src/curves/models/twisted_edwards_extended/mod.rs b/algebra/src/curves/models/twisted_edwards_extended/mod.rs
index 1ca2531a8..dd82860a4 100644
--- a/algebra/src/curves/models/twisted_edwards_extended/mod.rs
+++ b/algebra/src/curves/models/twisted_edwards_extended/mod.rs
@@ -1,5 +1,6 @@
 use rand::{Rng, distributions::{Standard, Distribution}};
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use std::{
     fmt::{Display, Formatter, Result as FmtResult},
     io::{Read, Result as IoResult, Write},
@@ -105,23 +106,25 @@ impl<P: Parameters> GroupAffine<P> {
     }
 }
 
+impl<P: Parameters> Zero for GroupAffine<P> {
+    fn zero() -> Self {
+        Self::new(P::BaseField::zero(), P::BaseField::one())
+    }
+
+    fn is_zero(&self) -> bool {
+        self.x.is_zero() & self.y.is_one()
+    }
+}
+
 impl<P: Parameters> AffineCurve for GroupAffine<P> {
     type BaseField = P::BaseField;
     type ScalarField = P::ScalarField;
     type Projective = GroupProjective<P>;
 
-    fn zero() -> Self {
-        Self::new(Self::BaseField::zero(), Self::BaseField::one())
-    }
-
     fn prime_subgroup_generator() -> Self {
         Self::new(P::AFFINE_GENERATOR_COEFFS.0, P::AFFINE_GENERATOR_COEFFS.1)
     }
 
-    fn is_zero(&self) -> bool {
-        self.x.is_zero() & self.y.is_one()
-    }
-
     fn mul<S: Into<<Self::ScalarField as PrimeField>::BigInt>>(&self, by: S) -> GroupProjective<P> {
         self.mul_bits(BitIterator::new(by.into()))
     }
@@ -147,6 +150,15 @@ impl<P: Parameters> Neg for GroupAffine<P> {
     }
 }
 
+impl<P: Parameters> Add<Self> for GroupAffine<P> {
+    type Output = Self;
+    fn add(self, other: Self) -> Self {
+        let mut copy = self;
+        copy += &other;
+        copy
+    }
+}
+
 impl<'a, P: Parameters> Add<&'a Self> for GroupAffine<P> {
     type Output = Self;
     fn add(self, other: &'a Self) -> Self {
@@ -188,11 +200,11 @@ impl<'a, P: Parameters> SubAssign<&'a Self> for GroupAffine<P> {
     }
 }
 
-impl<'a, P: Parameters> Mul<&'a P::ScalarField> for GroupAffine<P> {
+impl<P: Parameters> Mul<P::ScalarField> for GroupAffine<P> {
     type Output = Self;
-    fn mul(self, other: &'a P::ScalarField) -> Self {
+    fn mul(self, other: P::ScalarField) -> Self {
         let mut copy = self;
-        copy *= other;
+        copy *= &other;
         copy
     }
 }
@@ -247,13 +259,6 @@ mod group_impl {
 
     impl<P: Parameters> Group for GroupAffine<P> {
         type ScalarField = P::ScalarField;
-        fn zero() -> Self {
-            <Self as AffineCurve>::zero()
-        }
-
-        fn is_zero(&self) -> bool {
-            <Self as AffineCurve>::is_zero(&self)
-        }
 
         #[inline]
         #[must_use]
@@ -367,11 +372,7 @@ impl<P: Parameters> GroupProjective<P> {
     }
 }
 
-impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
-    type BaseField = P::BaseField;
-    type ScalarField = P::ScalarField;
-    type Affine = GroupAffine<P>;
-
+impl<P: Parameters> Zero for GroupProjective<P> {
     fn zero() -> Self {
         Self::new(
             P::BaseField::zero(),
@@ -381,13 +382,19 @@ impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
         )
     }
 
-    fn prime_subgroup_generator() -> Self {
-        GroupAffine::prime_subgroup_generator().into()
-    }
-
     fn is_zero(&self) -> bool {
         self.x.is_zero() && self.y == self.z && !self.y.is_zero() && self.t.is_zero()
     }
+}
+
+impl<P: Parameters> ProjectiveCurve for GroupProjective<P> {
+    type BaseField = P::BaseField;
+    type ScalarField = P::ScalarField;
+    type Affine = GroupAffine<P>;
+
+    fn prime_subgroup_generator() -> Self {
+        GroupAffine::prime_subgroup_generator().into()
+    }
 
     fn is_normalized(&self) -> bool {
         self.z.is_one()
@@ -511,11 +518,11 @@ impl<P: Parameters> Neg for GroupProjective<P> {
     }
 }
 
-impl<'a, P: Parameters> Add<&'a Self> for GroupProjective<P> {
+impl<P: Parameters> Add<Self> for GroupProjective<P> {
     type Output = Self;
-    fn add(self, other: &'a Self) -> Self {
+    fn add(self, other: Self) -> Self {
         let mut copy = self;
-        copy += other;
+        copy += &other;
         copy
     }
 }
@@ -652,4 +659,3 @@ impl<P: MontgomeryParameters> MontgomeryGroupAffine<P> {
         }
     }
 }
-
diff --git a/algebra/src/curves/models/twisted_edwards_extended/tests.rs b/algebra/src/curves/models/twisted_edwards_extended/tests.rs
index 148d82db2..8817aa1b9 100644
--- a/algebra/src/curves/models/twisted_edwards_extended/tests.rs
+++ b/algebra/src/curves/models/twisted_edwards_extended/tests.rs
@@ -1,4 +1,5 @@
 use crate::{fields::Field, TEModelParameters, MontgomeryModelParameters};
+use num_traits::One;
 
 pub(crate) fn montgomery_conversion_test<P>()
     where
diff --git a/algebra/src/curves/sw6/mod.rs b/algebra/src/curves/sw6/mod.rs
index aa4a673bd..38f861de3 100644
--- a/algebra/src/curves/sw6/mod.rs
+++ b/algebra/src/curves/sw6/mod.rs
@@ -10,6 +10,7 @@ use crate::{
         BitIterator, Field, FpParameters,
     },
 };
+use num_traits::One;
 
 pub mod g1;
 pub use self::g1::{G1Affine, G1Projective};
diff --git a/algebra/src/curves/sw6/tests.rs b/algebra/src/curves/sw6/tests.rs
index ad15d74cc..7dc4dcd69 100644
--- a/algebra/src/curves/sw6/tests.rs
+++ b/algebra/src/curves/sw6/tests.rs
@@ -6,6 +6,7 @@ use crate::{
     },
     groups::tests::group_test,
 };
+use num_traits::One;
 
 #[test]
 fn test_g1_projective_curve() {
diff --git a/algebra/src/curves/tests.rs b/algebra/src/curves/tests.rs
index da555165d..fcf3c5a27 100644
--- a/algebra/src/curves/tests.rs
+++ b/algebra/src/curves/tests.rs
@@ -1,8 +1,9 @@
 use crate::{
     curves::{AffineCurve, ProjectiveCurve},
-    fields::{Field, PrimeField},
+    fields::PrimeField,
 };
 use crate::UniformRand;
+use num_traits::Zero;
 use rand::SeedableRng;
 use rand_xorshift::XorShiftRng;
 
diff --git a/algebra/src/fields/bls12_377/fq6.rs b/algebra/src/fields/bls12_377/fq6.rs
index 366b1ef2f..5eec974ca 100644
--- a/algebra/src/fields/bls12_377/fq6.rs
+++ b/algebra/src/fields/bls12_377/fq6.rs
@@ -193,8 +193,9 @@ impl Fp6Parameters for Fq6Parameters {
 #[cfg(test)]
 mod test {
     use super::*;
-    use crate::fields::Field;
+    
     use crate::UniformRand;
+    use num_traits::{One, Zero};
 use rand::SeedableRng;
 use rand_xorshift::XorShiftRng;
 
diff --git a/algebra/src/fields/bls12_377/tests.rs b/algebra/src/fields/bls12_377/tests.rs
index e519e710b..2bc8725b8 100644
--- a/algebra/src/fields/bls12_377/tests.rs
+++ b/algebra/src/fields/bls12_377/tests.rs
@@ -8,6 +8,7 @@ use crate::{
     },
 };
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use rand::SeedableRng;
 use rand_xorshift::XorShiftRng;
 use std::{
diff --git a/algebra/src/fields/bls12_381/fq12.rs b/algebra/src/fields/bls12_381/fq12.rs
index fd32a2c18..408818910 100644
--- a/algebra/src/fields/bls12_381/fq12.rs
+++ b/algebra/src/fields/bls12_381/fq12.rs
@@ -210,9 +210,9 @@ mod test {
     use super::*;
     use crate::fields::{
         bls12_381::{fq2::Fq2, fq6::Fq6},
-        Field,
     };
     use crate::UniformRand;
+    use num_traits::{One, Zero};
 use rand::SeedableRng;
 use rand_xorshift::XorShiftRng;
 
diff --git a/algebra/src/fields/bls12_381/tests.rs b/algebra/src/fields/bls12_381/tests.rs
index 7b946ac84..c128fe062 100644
--- a/algebra/src/fields/bls12_381/tests.rs
+++ b/algebra/src/fields/bls12_381/tests.rs
@@ -11,6 +11,7 @@ use crate::{
     },
 };
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use rand::SeedableRng;
 use rand_xorshift::XorShiftRng;
 use std::{
diff --git a/algebra/src/fields/jubjub/tests.rs b/algebra/src/fields/jubjub/tests.rs
index 7212175a7..f314e2a3c 100644
--- a/algebra/src/fields/jubjub/tests.rs
+++ b/algebra/src/fields/jubjub/tests.rs
@@ -9,6 +9,7 @@ use crate::{
         PrimeField, SquareRootField,
     },
 };
+use num_traits::{One, Zero};
 use std::str::FromStr;
 
 #[test]
diff --git a/algebra/src/fields/mod.rs b/algebra/src/fields/mod.rs
index e44bddf91..9c1c65c5a 100644
--- a/algebra/src/fields/mod.rs
+++ b/algebra/src/fields/mod.rs
@@ -3,6 +3,7 @@ use crate::{
     bytes::{FromBytes, ToBytes},
     UniformRand,
 };
+use num_traits::{One, Zero};
 use std::{
     fmt::{Debug, Display},
     hash::Hash,
@@ -64,9 +65,11 @@ pub trait Field:
     + Sync
     + 'static
     + Eq
+    + One
     + Ord
     + Neg<Output = Self>
     + UniformRand
+    + Zero
     + Sized
     + Hash
     + From<u128>
@@ -75,26 +78,16 @@ pub trait Field:
     + From<u16>
     + From<u8>
     + for<'a> Add<&'a Self, Output = Self>
+    + Add<Self, Output = Self>
     + for<'a> Sub<&'a Self, Output = Self>
     + for<'a> Mul<&'a Self, Output = Self>
+    + Mul<Self, Output = Self>
     + for<'a> Div<&'a Self, Output = Self>
     + for<'a> AddAssign<&'a Self>
     + for<'a> SubAssign<&'a Self>
     + for<'a> MulAssign<&'a Self>
     + for<'a> DivAssign<&'a Self>
 {
-    /// Returns the zero element of the field, the additive identity.
-    fn zero() -> Self;
-
-    /// Returns true if and only if `self == Self::zero()`.
-    fn is_zero(&self) -> bool;
-
-    /// Returns the one element of the field, a field generator.
-    fn one() -> Self;
-
-    /// Returns true if and only if `self == Self::one()`.
-    fn is_one(&self) -> bool;
-
     /// Returns the characteristic of the field.
     fn characteristic<'a>() -> &'a [u64];
 
diff --git a/algebra/src/fields/models/fp12_2over3over2.rs b/algebra/src/fields/models/fp12_2over3over2.rs
index 1ab1f9147..f396ff5ec 100644
--- a/algebra/src/fields/models/fp12_2over3over2.rs
+++ b/algebra/src/fields/models/fp12_2over3over2.rs
@@ -1,5 +1,6 @@
 use rand::{Rng, distributions::{Standard, Distribution}};
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use std::{
     cmp::Ordering,
     io::{Read, Result as IoResult, Write},
@@ -216,23 +217,26 @@ impl<P: Fp12Parameters> Distribution<Fp12<P>> for Standard {
     }
 }
 
-impl<P: Fp12Parameters> Field for Fp12<P> {
+impl<P: Fp12Parameters> Zero for Fp12<P> {
     fn zero() -> Self {
         Self::new(Fp6::zero(), Fp6::zero())
     }
 
-    fn one() -> Self {
-        Self::new(Fp6::one(), Fp6::zero())
-    }
-
     fn is_zero(&self) -> bool {
         self.c0.is_zero() && self.c1.is_zero()
     }
+}
 
+impl<P: Fp12Parameters> One for Fp12<P> {
+    fn one() -> Self {
+        Self::new(Fp6::one(), Fp6::zero())
+    }
     fn is_one(&self) -> bool {
         self.c0.is_one() && self.c1.is_zero()
     }
+}
 
+impl<P: Fp12Parameters> Field for Fp12<P> {
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         Fp6::<P::Fp6Params>::characteristic()
diff --git a/algebra/src/fields/models/fp2.rs b/algebra/src/fields/models/fp2.rs
index 9b6b953ef..446ebe44d 100644
--- a/algebra/src/fields/models/fp2.rs
+++ b/algebra/src/fields/models/fp2.rs
@@ -1,4 +1,5 @@
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use rand::{
     distributions::{Distribution, Standard},
     Rng,
@@ -73,7 +74,7 @@ impl<P: Fp2Parameters> Fp2<P> {
     }
 }
 
-impl<P: Fp2Parameters> Field for Fp2<P> {
+impl<P: Fp2Parameters> Zero for Fp2<P> {
     fn zero() -> Self {
         Fp2::new(P::Fp::zero(), P::Fp::zero())
     }
@@ -81,7 +82,9 @@ impl<P: Fp2Parameters> Field for Fp2<P> {
     fn is_zero(&self) -> bool {
         self.c0.is_zero() && self.c1.is_zero()
     }
+}
 
+impl<P: Fp2Parameters> One for Fp2<P> {
     fn one() -> Self {
         Fp2::new(P::Fp::one(), P::Fp::zero())
     }
@@ -89,7 +92,9 @@ impl<P: Fp2Parameters> Field for Fp2<P> {
     fn is_one(&self) -> bool {
         self.c0.is_one() && self.c1.is_zero()
     }
+}
 
+impl<P: Fp2Parameters> Field for Fp2<P> {
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         P::Fp::characteristic()
diff --git a/algebra/src/fields/models/fp3.rs b/algebra/src/fields/models/fp3.rs
index 7f7b87fc2..dfbaef361 100644
--- a/algebra/src/fields/models/fp3.rs
+++ b/algebra/src/fields/models/fp3.rs
@@ -1,4 +1,5 @@
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use rand::{
     distributions::{Distribution, Standard},
     Rng,
@@ -91,7 +92,7 @@ impl<P: Fp3Parameters> Fp3<P> {
     }
 }
 
-impl<P: Fp3Parameters> Field for Fp3<P> {
+impl<P: Fp3Parameters> Zero for Fp3<P> {
     fn zero() -> Self {
         Fp3 {
             c0:          P::Fp::zero(),
@@ -104,7 +105,9 @@ impl<P: Fp3Parameters> Field for Fp3<P> {
     fn is_zero(&self) -> bool {
         self.c0.is_zero() && self.c1.is_zero() && self.c2.is_zero()
     }
+}
 
+impl<P: Fp3Parameters> One for Fp3<P> {
     fn one() -> Self {
         Fp3 {
             c0:          P::Fp::one(),
@@ -117,7 +120,9 @@ impl<P: Fp3Parameters> Field for Fp3<P> {
     fn is_one(&self) -> bool {
         self.c0.is_one() && self.c1.is_zero() && self.c2.is_zero()
     }
+}
 
+impl<P: Fp3Parameters> Field for Fp3<P> {
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         P::Fp::characteristic()
diff --git a/algebra/src/fields/models/fp6_2over3.rs b/algebra/src/fields/models/fp6_2over3.rs
index bf8962136..ac8a7b4ee 100644
--- a/algebra/src/fields/models/fp6_2over3.rs
+++ b/algebra/src/fields/models/fp6_2over3.rs
@@ -1,4 +1,5 @@
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use rand::{
     distributions::{Distribution, Standard},
     Rng,
@@ -99,7 +100,7 @@ impl<P: Fp6Parameters> Fp6<P> {
     }
 }
 
-impl<P: Fp6Parameters> Field for Fp6<P> {
+impl<P: Fp6Parameters> Zero for Fp6<P> {
     fn zero() -> Self {
         Fp6 {
             c0:          Fp3::zero(),
@@ -111,7 +112,9 @@ impl<P: Fp6Parameters> Field for Fp6<P> {
     fn is_zero(&self) -> bool {
         self.c0.is_zero() && self.c1.is_zero()
     }
+}
 
+impl<P: Fp6Parameters> One for Fp6<P> {
     fn one() -> Self {
         Fp6 {
             c0:          Fp3::one(),
@@ -123,7 +126,9 @@ impl<P: Fp6Parameters> Field for Fp6<P> {
     fn is_one(&self) -> bool {
         self.c0.is_one() && self.c1.is_zero()
     }
+}
 
+impl<P: Fp6Parameters> Field for Fp6<P> {
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         Fp3::<P::Fp3Params>::characteristic()
diff --git a/algebra/src/fields/models/fp6_3over2.rs b/algebra/src/fields/models/fp6_3over2.rs
index 119063d6e..f54c0f16f 100644
--- a/algebra/src/fields/models/fp6_3over2.rs
+++ b/algebra/src/fields/models/fp6_3over2.rs
@@ -1,4 +1,5 @@
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use rand::{
     distributions::{Distribution, Standard},
     Rng,
@@ -144,7 +145,7 @@ impl<P: Fp6Parameters> Fp6<P> {
     }
 }
 
-impl<P: Fp6Parameters> Field for Fp6<P> {
+impl<P: Fp6Parameters> Zero for Fp6<P> {
     fn zero() -> Self {
         Self::new(Fp2::zero(), Fp2::zero(), Fp2::zero())
     }
@@ -152,7 +153,9 @@ impl<P: Fp6Parameters> Field for Fp6<P> {
     fn is_zero(&self) -> bool {
         self.c0.is_zero() && self.c1.is_zero() && self.c2.is_zero()
     }
+}
 
+impl<P: Fp6Parameters> One for Fp6<P> {
     fn one() -> Self {
         Self::new(Fp2::one(), Fp2::zero(), Fp2::zero())
     }
@@ -160,7 +163,9 @@ impl<P: Fp6Parameters> Field for Fp6<P> {
     fn is_one(&self) -> bool {
         self.c0.is_one() && self.c1.is_zero() && self.c2.is_zero()
     }
+}
 
+impl<P: Fp6Parameters> Field for Fp6<P> {
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         Fp2::<P::Fp2Params>::characteristic()
diff --git a/algebra/src/fields/models/fp_256.rs b/algebra/src/fields/models/fp_256.rs
index f47593f1d..2f8a379fc 100644
--- a/algebra/src/fields/models/fp_256.rs
+++ b/algebra/src/fields/models/fp_256.rs
@@ -1,3 +1,4 @@
+use num_traits::{One, Zero};
 use std::{
     cmp::{Ord, Ordering, PartialOrd},
     fmt::{Display, Formatter, Result as FmtResult},
@@ -105,7 +106,7 @@ impl<P: Fp256Parameters> Fp256<P> {
     }
 }
 
-impl<P: Fp256Parameters> Field for Fp256<P> {
+impl<P: Fp256Parameters> Zero for Fp256<P> {
     #[inline]
     fn zero() -> Self {
         Fp256::<P>(BigInteger::from(0), PhantomData)
@@ -115,7 +116,21 @@ impl<P: Fp256Parameters> Field for Fp256<P> {
     fn is_zero(&self) -> bool {
         self.0.is_zero()
     }
+}
+
+impl<P: Fp256Parameters> One for Fp256<P> {
+    #[inline]
+    fn one() -> Self {
+        Fp256::<P>(P::R, PhantomData)
+    }
+
+    #[inline]
+    fn is_one(&self) -> bool {
+        self == &Self::one()
+    }
+}
 
+impl<P: Fp256Parameters> Field for Fp256<P> {
     #[inline]
     fn double(&self) -> Self {
         let mut temp = *self;
@@ -132,16 +147,6 @@ impl<P: Fp256Parameters> Field for Fp256<P> {
         self
     }
 
-    #[inline]
-    fn one() -> Self {
-        Fp256::<P>(P::R, PhantomData)
-    }
-
-    #[inline]
-    fn is_one(&self) -> bool {
-        self == &Self::one()
-    }
-
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         P::MODULUS.as_ref()
diff --git a/algebra/src/fields/models/fp_320.rs b/algebra/src/fields/models/fp_320.rs
index 28c3956bc..fcd06b14f 100644
--- a/algebra/src/fields/models/fp_320.rs
+++ b/algebra/src/fields/models/fp_320.rs
@@ -1,3 +1,4 @@
+use num_traits::{One, Zero};
 use std::{
     cmp::{Ord, Ordering, PartialOrd},
     fmt::{Display, Formatter, Result as FmtResult},
@@ -121,7 +122,7 @@ impl<P: Fp320Parameters> Fp320<P> {
     }
 }
 
-impl<P: Fp320Parameters> Field for Fp320<P> {
+impl<P: Fp320Parameters> Zero for Fp320<P> {
     #[inline]
     fn zero() -> Self {
         Fp320::<P>(BigInteger::from(0), PhantomData)
@@ -131,7 +132,21 @@ impl<P: Fp320Parameters> Field for Fp320<P> {
     fn is_zero(&self) -> bool {
         self.0.is_zero()
     }
+}
+
+impl<P: Fp320Parameters> One for Fp320<P> {
+    #[inline]
+    fn one() -> Self {
+        Fp320::<P>(P::R, PhantomData)
+    }
+
+    #[inline]
+    fn is_one(&self) -> bool {
+        self.0 == P::R
+    }
+}
 
+impl<P: Fp320Parameters> Field for Fp320<P> {
     #[inline]
     fn double(&self) -> Self {
         let mut temp = *self;
@@ -148,16 +163,6 @@ impl<P: Fp320Parameters> Field for Fp320<P> {
         self
     }
 
-    #[inline]
-    fn one() -> Self {
-        Fp320::<P>(P::R, PhantomData)
-    }
-
-    #[inline]
-    fn is_one(&self) -> bool {
-        self.0 == P::R
-    }
-
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         P::MODULUS.as_ref()
diff --git a/algebra/src/fields/models/fp_384.rs b/algebra/src/fields/models/fp_384.rs
index 7191ffdc2..aa657e09c 100644
--- a/algebra/src/fields/models/fp_384.rs
+++ b/algebra/src/fields/models/fp_384.rs
@@ -1,3 +1,4 @@
+use num_traits::{One, Zero};
 use std::{
     cmp::{Ord, Ordering, PartialOrd},
     fmt::{Display, Formatter, Result as FmtResult},
@@ -139,7 +140,7 @@ impl<P: Fp384Parameters> Fp384<P> {
     }
 }
 
-impl<P: Fp384Parameters> Field for Fp384<P> {
+impl<P: Fp384Parameters> Zero for Fp384<P> {
     #[inline]
     fn zero() -> Self {
         Fp384::<P>(BigInteger::from(0), PhantomData)
@@ -149,7 +150,21 @@ impl<P: Fp384Parameters> Field for Fp384<P> {
     fn is_zero(&self) -> bool {
         self.0.is_zero()
     }
+}
+
+impl<P: Fp384Parameters> One for Fp384<P> {
+    #[inline]
+    fn one() -> Self {
+        Fp384::<P>(P::R, PhantomData)
+    }
+
+    #[inline]
+    fn is_one(&self) -> bool {
+        self.0 == P::R
+    }
+}
 
+impl<P: Fp384Parameters> Field for Fp384<P> {
     #[inline]
     fn double(&self) -> Self {
         let mut temp = *self;
@@ -166,16 +181,6 @@ impl<P: Fp384Parameters> Field for Fp384<P> {
         self
     }
 
-    #[inline]
-    fn one() -> Self {
-        Fp384::<P>(P::R, PhantomData)
-    }
-
-    #[inline]
-    fn is_one(&self) -> bool {
-        self.0 == P::R
-    }
-
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         P::MODULUS.as_ref()
diff --git a/algebra/src/fields/models/fp_768.rs b/algebra/src/fields/models/fp_768.rs
index ea7a44b00..ccd97b631 100644
--- a/algebra/src/fields/models/fp_768.rs
+++ b/algebra/src/fields/models/fp_768.rs
@@ -1,3 +1,4 @@
+use num_traits::{One, Zero};
 use std::{
     cmp::{Ord, Ordering, PartialOrd},
     fmt::{Display, Formatter, Result as FmtResult},
@@ -284,7 +285,7 @@ impl<P: Fp768Parameters> Fp768<P> {
     }
 }
 
-impl<P: Fp768Parameters> Field for Fp768<P> {
+impl<P: Fp768Parameters> Zero for Fp768<P> {
     #[inline]
     fn zero() -> Self {
         Fp768::<P>(BigInteger::from(0), PhantomData)
@@ -294,7 +295,21 @@ impl<P: Fp768Parameters> Field for Fp768<P> {
     fn is_zero(&self) -> bool {
         self.0.is_zero()
     }
+}
+
+impl<P: Fp768Parameters> One for Fp768<P> {
+    #[inline]
+    fn one() -> Self {
+        Fp768::<P>(P::R, PhantomData)
+    }
+
+    #[inline]
+    fn is_one(&self) -> bool {
+        self.0 == P::R
+    }
+}
 
+impl<P: Fp768Parameters> Field for Fp768<P> {
     #[inline]
     fn double(&self) -> Self {
         let mut temp = *self;
@@ -311,16 +326,6 @@ impl<P: Fp768Parameters> Field for Fp768<P> {
         self
     }
 
-    #[inline]
-    fn one() -> Self {
-        Fp768::<P>(P::R, PhantomData)
-    }
-
-    #[inline]
-    fn is_one(&self) -> bool {
-        self.0 == P::R
-    }
-
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         P::MODULUS.as_ref()
diff --git a/algebra/src/fields/models/fp_832.rs b/algebra/src/fields/models/fp_832.rs
index eb4ab2b08..f960b7969 100644
--- a/algebra/src/fields/models/fp_832.rs
+++ b/algebra/src/fields/models/fp_832.rs
@@ -3,6 +3,7 @@ use crate::{
     bytes::{FromBytes, ToBytes},
     fields::{Field, FpParameters, LegendreSymbol, PrimeField, SquareRootField},
 };
+use num_traits::{One, Zero};
 use std::{
     cmp::{Ord, Ordering, PartialOrd},
     fmt::{Display, Formatter, Result as FmtResult},
@@ -315,7 +316,7 @@ impl<P: Fp832Parameters> Fp832<P> {
     }
 }
 
-impl<P: Fp832Parameters> Field for Fp832<P> {
+impl<P: Fp832Parameters> Zero for Fp832<P> {
     #[inline]
     fn zero() -> Self {
         Fp832::<P>(BigInteger::from(0), PhantomData)
@@ -325,7 +326,21 @@ impl<P: Fp832Parameters> Field for Fp832<P> {
     fn is_zero(&self) -> bool {
         self.0.is_zero()
     }
+}
+
+impl<P: Fp832Parameters> One for Fp832<P> {
+    #[inline]
+    fn one() -> Self {
+        Fp832::<P>(P::R, PhantomData)
+    }
+
+    #[inline]
+    fn is_one(&self) -> bool {
+        self.0 == P::R
+    }
+}
 
+impl<P: Fp832Parameters> Field for Fp832<P> {
     #[inline]
     fn double(&self) -> Self {
         let mut temp = *self;
@@ -342,16 +357,6 @@ impl<P: Fp832Parameters> Field for Fp832<P> {
         self
     }
 
-    #[inline]
-    fn one() -> Self {
-        Fp832::<P>(P::R, PhantomData)
-    }
-
-    #[inline]
-    fn is_one(&self) -> bool {
-        self.0 == P::R
-    }
-
     #[inline]
     fn characteristic<'a>() -> &'a [u64] {
         P::MODULUS.as_ref()
diff --git a/algebra/src/groups/mod.rs b/algebra/src/groups/mod.rs
index 0004fa589..883014c0f 100644
--- a/algebra/src/groups/mod.rs
+++ b/algebra/src/groups/mod.rs
@@ -1,5 +1,6 @@
 use crate::BitIterator;
 use crate::UniformRand;
+use num_traits::Zero;
 use std::{
     fmt::{Debug, Display},
     hash::Hash,
@@ -29,19 +30,14 @@ pub trait Group:
     + Hash
     + Neg<Output = Self>
     + UniformRand
-    + for<'a> Add<&'a Self, Output = Self>
+    + Zero
+    + Add<Self, Output = Self>
     + for<'a> Sub<&'a Self, Output = Self>
     + for<'a> AddAssign<&'a Self>
     + for<'a> SubAssign<&'a Self>
 {
     type ScalarField: PrimeField + Into<<Self::ScalarField as PrimeField>::BigInt>;
 
-    /// Returns the additive identity.
-    fn zero() -> Self;
-
-    /// Returns `self == zero`.
-    fn is_zero(&self) -> bool;
-
     /// Returns `self + self`.
     #[must_use]
     fn double(&self) -> Self;
diff --git a/algebra/src/groups/tests.rs b/algebra/src/groups/tests.rs
index 38c548ce2..1c1cf13d1 100644
--- a/algebra/src/groups/tests.rs
+++ b/algebra/src/groups/tests.rs
@@ -1,6 +1,6 @@
 use super::Group;
-use crate::fields::Field;
 use crate::UniformRand;
+use num_traits::{One, Zero};
 use rand::SeedableRng;
 use rand_xorshift::XorShiftRng;
 
diff --git a/algebra/src/msm/variable_base.rs b/algebra/src/msm/variable_base.rs
index 8fd72395d..f1dac526b 100644
--- a/algebra/src/msm/variable_base.rs
+++ b/algebra/src/msm/variable_base.rs
@@ -1,8 +1,8 @@
 use crate::{
-    AffineCurve, BigInteger, Field, FpParameters, PrimeField,
+    AffineCurve, BigInteger, FpParameters, PrimeField,
     ProjectiveCurve,
 };
-#[cfg(feature = "parallel")]
+use num_traits::{One, Zero};
 use rayon::prelude::*;
 
 pub struct VariableBaseMSM;
@@ -84,7 +84,7 @@ impl VariableBaseMSM {
                 total.double_in_place();
             }
             total
-        }) + lowest
+        }) + *lowest
     }
 
     pub fn multi_scalar_mul<G: AffineCurve>(
@@ -135,7 +135,6 @@ mod test {
         assert_eq!(naive.into_affine(), fast.into_affine());
     }
 
-
     #[test]
     fn test_with_bls12_unequal_numbers() {
         const SAMPLES: usize = 1 << 10;
diff --git a/crypto-primitives/Cargo.toml b/crypto-primitives/Cargo.toml
index 3b937c80b..4d3566d5b 100644
--- a/crypto-primitives/Cargo.toml
+++ b/crypto-primitives/Cargo.toml
@@ -32,6 +32,7 @@ bench-utils = { path = "../bench-utils" }
 digest = "0.7"
 blake2 = "0.7"
 
+num-traits = { version = "0.2.11" }
 rand = { version = "0.7" }
 derivative = "1"
 rayon = "1"
diff --git a/crypto-primitives/src/merkle_tree/mod.rs b/crypto-primitives/src/merkle_tree/mod.rs
index df65dd07d..ea1f8dfe8 100644
--- a/crypto-primitives/src/merkle_tree/mod.rs
+++ b/crypto-primitives/src/merkle_tree/mod.rs
@@ -372,6 +372,7 @@ mod test {
         merkle_tree::*,
     };
     use algebra::curves::jubjub::JubJubAffine as JubJub;
+    use num_traits::Zero;
     use rand::SeedableRng;
     use rand_xorshift::XorShiftRng;
 
@@ -419,7 +420,6 @@ mod test {
     }
 
     fn bad_merkle_tree_verify<L: ToBytes + Clone + Eq>(leaves: &[L]) -> () {
-        use algebra::groups::Group;
         let mut rng = XorShiftRng::seed_from_u64(13423423u64);
 
         let crh_parameters = Rc::new(H::setup(&mut rng).unwrap());
diff --git a/crypto-primitives/src/nizk/mod.rs b/crypto-primitives/src/nizk/mod.rs
index 048c51c7f..1c30bc767 100644
--- a/crypto-primitives/src/nizk/mod.rs
+++ b/crypto-primitives/src/nizk/mod.rs
@@ -59,7 +59,8 @@ mod test {
     #[test]
     fn test_gm17() {
         use crate::nizk::{gm17::Gm17, NIZK};
-        use algebra::{curves::bls12_381::Bls12_381, fields::bls12_381::Fr, Field};
+        use algebra::{curves::bls12_381::Bls12_381, fields::bls12_381::Fr};
+        use num_traits::One;
         use r1cs_core::{ConstraintSynthesizer, ConstraintSystem, SynthesisError};
 
         #[derive(Copy, Clone)]
diff --git a/crypto-primitives/src/signature/schnorr/mod.rs b/crypto-primitives/src/signature/schnorr/mod.rs
index 29a8c2581..9e03e0599 100644
--- a/crypto-primitives/src/signature/schnorr/mod.rs
+++ b/crypto-primitives/src/signature/schnorr/mod.rs
@@ -6,6 +6,7 @@ use algebra::{
     to_bytes, ToConstraintField, UniformRand,
 };
 use digest::Digest;
+use num_traits::{One, Zero};
 use rand::Rng;
 use std::{
     hash::Hash,
diff --git a/ff-fft/Cargo.toml b/ff-fft/Cargo.toml
index 4fbe727b3..48b1696c0 100644
--- a/ff-fft/Cargo.toml
+++ b/ff-fft/Cargo.toml
@@ -20,6 +20,7 @@ license = "MIT/Apache-2.0"
 edition = "2018"
 
 [dependencies]
+num-traits = { version = "0.2.11" }
 rand = { version = "0.7" }
 algebra = { path = "../algebra", features = [ "parallel" ] }
 rayon = { version = "1" }
diff --git a/ff-fft/src/domain.rs b/ff-fft/src/domain.rs
index 83a628558..e47ad6a97 100644
--- a/ff-fft/src/domain.rs
+++ b/ff-fft/src/domain.rs
@@ -42,7 +42,6 @@ impl<F: PrimeField> fmt::Debug for EvaluationDomain<F> {
     fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
         write!(f, "Multiplicative subgroup of size {}", self.size)
     }
-
 }
 
 impl<F: PrimeField> EvaluationDomain<F> {
@@ -59,7 +58,6 @@ impl<F: PrimeField> EvaluationDomain<F> {
         t
     }
 
-
     /// Construct a domain that is large enough for evaluations of a polynomial
     /// having `num_coeffs` coefficients.
     pub fn new(num_coeffs: usize) -> Option<Self> {
@@ -442,6 +440,7 @@ mod tests {
     use algebra::Field;
     use algebra::fields::bls12_381::fr::Fr;
     use rand::{Rng, thread_rng};
+    use num_traits::Zero;
 
     #[test]
     fn vanishing_polynomial_evaluation() {
diff --git a/ff-fft/src/polynomial/dense.rs b/ff-fft/src/polynomial/dense.rs
index 9803e45f1..4f9ccd693 100644
--- a/ff-fft/src/polynomial/dense.rs
+++ b/ff-fft/src/polynomial/dense.rs
@@ -181,7 +181,6 @@ impl<'a, 'b, F: Field> AddAssign<&'a DensePolynomial<F>> for DensePolynomial<F>
             self.coeffs.truncate(0);
             self.coeffs.extend_from_slice(&other.coeffs);
         } else if other.is_zero() {
-
         } else if self.degree() >= other.degree() {
             for (a, b) in self.coeffs.iter_mut().zip(&other.coeffs) {
                 *a += b
@@ -298,7 +297,6 @@ impl<'a, 'b, F: Field> SubAssign<&'a DensePolynomial<F>> for DensePolynomial<F>
                 self.coeffs[i] -= coeff;
             }
         } else if other.is_zero() {
-
         } else if self.degree() >= other.degree() {
             for (a, b) in self.coeffs.iter_mut().zip(&other.coeffs) {
                 *a -= b
@@ -351,6 +349,7 @@ mod tests {
     use crate::polynomial::*;
     use algebra::fields::{bls12_381::fr::Fr, Field};
     use algebra::UniformRand; 
+    use num_traits::{One, Zero};
     use rand::thread_rng;
 
     #[test]
diff --git a/ff-fft/src/polynomial/mod.rs b/ff-fft/src/polynomial/mod.rs
index 5465ab92d..530b34917 100644
--- a/ff-fft/src/polynomial/mod.rs
+++ b/ff-fft/src/polynomial/mod.rs
@@ -34,7 +34,6 @@ impl<'a, F: 'a + Field> From<&'a DensePolynomial<F>> for DenseOrSparsePolynomial
     }
 }
 
-
 impl<F: Field> From<SparsePolynomial<F>> for DenseOrSparsePolynomial<'_, F> {
     fn from(other: SparsePolynomial<F>) -> Self {
         SPolynomial(Cow::Owned(other))
@@ -47,7 +46,6 @@ impl<'a, F: Field> From<&'a SparsePolynomial<F>> for DenseOrSparsePolynomial<'a,
     }
 }
 
-
 impl<F: Field> Into<DensePolynomial<F>> for DenseOrSparsePolynomial<'_, F> {
     fn into(self) -> DensePolynomial<F> {
         match self {
diff --git a/ff-fft/src/polynomial/sparse.rs b/ff-fft/src/polynomial/sparse.rs
index 776a763b1..3b75b4cc9 100644
--- a/ff-fft/src/polynomial/sparse.rs
+++ b/ff-fft/src/polynomial/sparse.rs
@@ -129,7 +129,7 @@ impl<F: Field> Into<DensePolynomial<F>> for SparsePolynomial<F> {
 mod tests {
     use crate::{EvaluationDomain, DensePolynomial, SparsePolynomial};
     use algebra::fields::bls12_381::fr::Fr;
-    use algebra::Field;
+    use num_traits::One;
 
     #[test]
     fn evaluate_over_domain() {
diff --git a/gm17/Cargo.toml b/gm17/Cargo.toml
index e2ff2261a..db63c6d22 100644
--- a/gm17/Cargo.toml
+++ b/gm17/Cargo.toml
@@ -27,6 +27,7 @@ ff-fft = { path = "../ff-fft" }
 r1cs-core = { path = "../r1cs-core" }
 bench-utils = { path = "../bench-utils" }
 
+num-traits = { version = "0.2.11" }
 rand = { version = "0.7" }
 rayon = { version = "1" }
 smallvec = { version = "0.6" }
diff --git a/gm17/examples/snark-scalability/gm17.rs b/gm17/examples/snark-scalability/gm17.rs
index 08cfa34b8..55ca79a7b 100644
--- a/gm17/examples/snark-scalability/gm17.rs
+++ b/gm17/examples/snark-scalability/gm17.rs
@@ -27,6 +27,7 @@
 
 use csv;
 
+use num_traits::One;
 // For randomness (during paramgen and proof generation)
 use rand::thread_rng;
 
@@ -39,7 +40,7 @@ use std::{
 // Bring in some tools for using pairing-friendly curves
 // We're going to use the BLS12-377 pairing-friendly elliptic curve.
 use algebra::curves::bls12_377::Bls12_377;
-use algebra::fields::{bls12_377::fr::Fr, Field};
+use algebra::fields::bls12_377::fr::Fr;
 
 // We're going to use the Groth-Maller 17 proving system.
 use gm17::{
diff --git a/gm17/src/generator.rs b/gm17/src/generator.rs
index dd1010d14..0500e9860 100644
--- a/gm17/src/generator.rs
+++ b/gm17/src/generator.rs
@@ -4,6 +4,7 @@ use algebra::{
     AffineCurve, Field, PairingEngine, PrimeField, ProjectiveCurve,
 };
 
+use num_traits::{One, Zero};
 use rand::Rng;
 use rayon::prelude::*;
 use r1cs_core::{ConstraintSynthesizer, ConstraintSystem, Index, LinearCombination, SynthesisError, Variable};
@@ -298,8 +299,6 @@ where
     );
     end_timer!(b_time);
 
-
-
     end_timer!(proving_key_time);
 
     // Generate R1CS verification key
diff --git a/gm17/src/prover.rs b/gm17/src/prover.rs
index 76e81c954..4a65ac696 100644
--- a/gm17/src/prover.rs
+++ b/gm17/src/prover.rs
@@ -2,7 +2,7 @@ use rand::Rng;
 use rayon::prelude::*;
 
 use algebra::{
-    UniformRand, msm::VariableBaseMSM, AffineCurve, Field, PairingEngine, PrimeField, ProjectiveCurve,
+    UniformRand, msm::VariableBaseMSM, AffineCurve, PairingEngine, PrimeField, ProjectiveCurve,
 };
 
 use crate::{Parameters, Proof};
@@ -10,6 +10,7 @@ use crate::r1cs_to_sap::R1CStoSAP;
 
 use r1cs_core::{ConstraintSynthesizer, ConstraintSystem, Index, LinearCombination, SynthesisError, Variable};
 
+use num_traits::{One, Zero};
 use smallvec::SmallVec;
 
 use std::{
diff --git a/gm17/src/r1cs_to_sap.rs b/gm17/src/r1cs_to_sap.rs
index bce87cb83..5c439880d 100644
--- a/gm17/src/r1cs_to_sap.rs
+++ b/gm17/src/r1cs_to_sap.rs
@@ -2,6 +2,7 @@ use ff_fft::EvaluationDomain;
 use algebra::{Field, PairingEngine};
 
 use crate::{generator::KeypairAssembly, prover::ProvingAssignment};
+use num_traits::{One, Zero};
 use r1cs_core::{Index, SynthesisError};
 
 use rayon::prelude::*;
@@ -227,7 +228,7 @@ impl R1CStoSAP {
             tmp.double_in_place();
 
             let assignment = full_input_assignment[extra_var_offset2 + i];
-            c[extra_constr_offset + 2 * i - 1] = tmp + &assignment;
+            c[extra_constr_offset + 2 * i - 1] = tmp + assignment;
             c[extra_constr_offset + 2 * i] = assignment;
         }
 
diff --git a/gm17/src/test.rs b/gm17/src/test.rs
index f3d796f84..f10894666 100644
--- a/gm17/src/test.rs
+++ b/gm17/src/test.rs
@@ -1,5 +1,7 @@
-use algebra::Field;
+use algebra::fields::Field;
+use num_traits::Zero;
 use r1cs_core::{ConstraintSynthesizer, ConstraintSystem, SynthesisError};
+
 struct MySillyCircuit<F: Field> {
     a: Option<F>,
     b: Option<F>,
@@ -79,7 +81,7 @@ mod sw6 {
 
     use rand::thread_rng;
 
-    use algebra::{UniformRand, curves::sw6::SW6, fields::sw6::Fr as SW6Fr, Field};
+    use algebra::{UniformRand, curves::sw6::SW6, fields::sw6::Fr as SW6Fr};
 
     #[test]
     fn prove_and_verify() {
diff --git a/gm17/src/verifier.rs b/gm17/src/verifier.rs
index dc853b927..900208e4b 100644
--- a/gm17/src/verifier.rs
+++ b/gm17/src/verifier.rs
@@ -1,4 +1,5 @@
-use algebra::{AffineCurve, Field, PairingCurve, PairingEngine, PrimeField, ProjectiveCurve};
+use algebra::{AffineCurve, PairingCurve, PairingEngine, PrimeField, ProjectiveCurve};
+use num_traits::One;
 
 use super::{PreparedVerifyingKey, Proof, VerifyingKey};
 
diff --git a/groth16/Cargo.toml b/groth16/Cargo.toml
index 717ae5529..fc1f1f735 100644
--- a/groth16/Cargo.toml
+++ b/groth16/Cargo.toml
@@ -28,6 +28,7 @@ ff-fft = { path = "../ff-fft" }
 r1cs-core = { path = "../r1cs-core" }
 bench-utils = { path = "../bench-utils" }
 
+num-traits = { version = "0.2.11" }
 rand = { version = "0.7" }
 rayon = { version = "1" }
 smallvec = { version = "0.6" }
diff --git a/groth16/examples/snark-scalability/groth16.rs b/groth16/examples/snark-scalability/groth16.rs
index 9caef8eda..ec06cacf4 100644
--- a/groth16/examples/snark-scalability/groth16.rs
+++ b/groth16/examples/snark-scalability/groth16.rs
@@ -40,7 +40,7 @@ use std::{
 // We're going to use the BLS12-377 pairing-friendly elliptic curve.
 use algebra::{
     curves::bls12_377::Bls12_377,
-    fields::{bls12_377::fr::Fr, Field},
+    fields::bls12_377::fr::Fr,
 };
 
 // We're going to use the Groth 16 proving system.
@@ -48,6 +48,7 @@ use groth16::{
     create_random_proof, generate_random_parameters, prepare_verifying_key, verify_proof,
 };
 
+use num_traits::One;
 use std::{env, fs::OpenOptions, path::PathBuf, process};
 
 mod constraints;
diff --git a/groth16/src/generator.rs b/groth16/src/generator.rs
index 6c217ebe5..65d96891f 100644
--- a/groth16/src/generator.rs
+++ b/groth16/src/generator.rs
@@ -4,6 +4,7 @@ use algebra::{
 };
 use ff_fft::EvaluationDomain;
 
+use num_traits::{One, Zero};
 use r1cs_core::{
     ConstraintSynthesizer, ConstraintSystem, Index, LinearCombination, SynthesisError, Variable,
 };
diff --git a/groth16/src/prover.rs b/groth16/src/prover.rs
index a9b847604..cd297c4ff 100644
--- a/groth16/src/prover.rs
+++ b/groth16/src/prover.rs
@@ -2,11 +2,12 @@ use rand::Rng;
 use rayon::prelude::*;
 
 use algebra::{
-    groups::Group, msm::VariableBaseMSM, AffineCurve, Field, PairingEngine, PrimeField,
+    groups::Group, msm::VariableBaseMSM, AffineCurve, PairingEngine, PrimeField,
     ProjectiveCurve, UniformRand,
 };
 
 use crate::{r1cs_to_qap::R1CStoQAP, Parameters, Proof};
+use num_traits::{One, Zero};
 
 use r1cs_core::{
     ConstraintSynthesizer, ConstraintSystem, Index, LinearCombination, SynthesisError, Variable,
diff --git a/groth16/src/r1cs_to_qap.rs b/groth16/src/r1cs_to_qap.rs
index ac95ab2b8..d5805be3c 100644
--- a/groth16/src/r1cs_to_qap.rs
+++ b/groth16/src/r1cs_to_qap.rs
@@ -1,7 +1,8 @@
-use algebra::{Field, PairingEngine};
+use algebra::PairingEngine;
 use ff_fft::EvaluationDomain;
 
 use crate::{generator::KeypairAssembly, prover::ProvingAssignment};
+use num_traits::{One, Zero};
 use r1cs_core::{Index, SynthesisError};
 
 use rayon::prelude::*;
diff --git a/groth16/src/test.rs b/groth16/src/test.rs
index 86bdc89c7..4d36df4bf 100644
--- a/groth16/src/test.rs
+++ b/groth16/src/test.rs
@@ -82,10 +82,10 @@ mod sw6 {
         create_random_proof, generate_random_parameters, prepare_verifying_key, verify_proof,
     };
 
+    use algebra::{curves::sw6::SW6, fields::sw6::Fr as SW6Fr, UniformRand};
+    use num_traits::Zero;
     use rand::thread_rng;
 
-    use algebra::{curves::sw6::SW6, fields::sw6::Fr as SW6Fr, Field, UniformRand};
-
     #[test]
     fn prove_and_verify() {
         let rng = &mut thread_rng();
diff --git a/r1cs-std/Cargo.toml b/r1cs-std/Cargo.toml
index a79739826..b72a8f055 100644
--- a/r1cs-std/Cargo.toml
+++ b/r1cs-std/Cargo.toml
@@ -25,6 +25,7 @@ edition = "2018"
 algebra = { path = "../algebra" }
 r1cs-core = { path = "../r1cs-core" }
 derivative = "1"
+num-traits = { version = "0.2.11" }
 radix_trie = "0.1"
 
 [dev-dependencies]
diff --git a/r1cs-std/src/bits/boolean.rs b/r1cs-std/src/bits/boolean.rs
index 0bcbacf6e..64f9f4402 100644
--- a/r1cs-std/src/bits/boolean.rs
+++ b/r1cs-std/src/bits/boolean.rs
@@ -832,13 +832,12 @@ impl<ConstraintF: Field> CondSelectGadget<ConstraintF> for Boolean {
     }
 }
 
-    
-
 #[cfg(test)]
 mod test {
     use super::{AllocatedBit, Boolean};
     use crate::{prelude::*, test_constraint_system::TestConstraintSystem};
     use algebra::{fields::bls12_381::Fr, BitIterator, Field, PrimeField, UniformRand};
+    use num_traits::{One, Zero};
     use r1cs_core::ConstraintSystem;
     use rand::SeedableRng;
     use rand_xorshift::XorShiftRng;
@@ -903,8 +902,8 @@ mod test {
                 assert_eq!(c.value.unwrap(), *a_val | *b_val);
 
                 assert!(cs.is_satisfied());
-                assert!(cs.get("a/boolean") == if *a_val { Field::one() } else { Field::zero() });
-                assert!(cs.get("b/boolean") == if *b_val { Field::one() } else { Field::zero() });
+                assert!(cs.get("a/boolean") == if *a_val { Fr::one() } else { Fr::zero() });
+                assert!(cs.get("b/boolean") == if *b_val { Fr::one() } else { Fr::zero() });
             }
         }
     }
@@ -920,14 +919,14 @@ mod test {
                 assert_eq!(c.value.unwrap(), *a_val & *b_val);
 
                 assert!(cs.is_satisfied());
-                assert!(cs.get("a/boolean") == if *a_val { Field::one() } else { Field::zero() });
-                assert!(cs.get("b/boolean") == if *b_val { Field::one() } else { Field::zero() });
+                assert!(cs.get("a/boolean") == if *a_val { Fr::one() } else { Fr::zero() });
+                assert!(cs.get("b/boolean") == if *b_val { Fr::one() } else { Fr::zero() });
                 assert!(
                     cs.get("and result")
                         == if *a_val & *b_val {
-                            Field::one()
+                            Fr::one()
                         } else {
-                            Field::zero()
+                            Fr::zero()
                         }
                 );
 
@@ -935,9 +934,9 @@ mod test {
                 cs.set(
                     "and result",
                     if *a_val & *b_val {
-                        Field::zero()
+                        Fr::zero()
                     } else {
-                        Field::one()
+                        Fr::one()
                     },
                 );
                 assert!(!cs.is_satisfied());
@@ -956,14 +955,14 @@ mod test {
                 assert_eq!(c.value.unwrap(), *a_val & !*b_val);
 
                 assert!(cs.is_satisfied());
-                assert!(cs.get("a/boolean") == if *a_val { Field::one() } else { Field::zero() });
-                assert!(cs.get("b/boolean") == if *b_val { Field::one() } else { Field::zero() });
+                assert!(cs.get("a/boolean") == if *a_val { Fr::one() } else { Fr::zero() });
+                assert!(cs.get("b/boolean") == if *b_val { Fr::one() } else { Fr::zero() });
                 assert!(
                     cs.get("and not result")
                         == if *a_val & !*b_val {
-                            Field::one()
+                            Fr::one()
                         } else {
-                            Field::zero()
+                            Fr::zero()
                         }
                 );
 
@@ -971,9 +970,9 @@ mod test {
                 cs.set(
                     "and not result",
                     if *a_val & !*b_val {
-                        Field::zero()
+                        Fr::zero()
                     } else {
-                        Field::one()
+                        Fr::one()
                     },
                 );
                 assert!(!cs.is_satisfied());
@@ -992,14 +991,14 @@ mod test {
                 assert_eq!(c.value.unwrap(), !*a_val & !*b_val);
 
                 assert!(cs.is_satisfied());
-                assert!(cs.get("a/boolean") == if *a_val { Field::one() } else { Field::zero() });
-                assert!(cs.get("b/boolean") == if *b_val { Field::one() } else { Field::zero() });
+                assert!(cs.get("a/boolean") == if *a_val { Fr::one() } else { Fr::zero() });
+                assert!(cs.get("b/boolean") == if *b_val { Fr::one() } else { Fr::zero() });
                 assert!(
                     cs.get("nor result")
                         == if !*a_val & !*b_val {
-                            Field::one()
+                            Fr::one()
                         } else {
-                            Field::zero()
+                            Fr::zero()
                         }
                 );
 
@@ -1007,9 +1006,9 @@ mod test {
                 cs.set(
                     "nor result",
                     if !*a_val & !*b_val {
-                        Field::zero()
+                        Fr::zero()
                     } else {
-                        Field::one()
+                        Fr::one()
                     },
                 );
                 assert!(!cs.is_satisfied());
@@ -1235,7 +1234,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1243,7 +1242,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1251,7 +1250,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1259,7 +1258,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
 
@@ -1270,7 +1269,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1278,7 +1277,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1286,7 +1285,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1294,7 +1293,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
 
@@ -1305,7 +1304,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1313,7 +1312,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1321,7 +1320,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1329,7 +1328,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
 
@@ -1340,7 +1339,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1348,7 +1347,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Not(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1356,7 +1355,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::one());
+                        assert!(cs.get("xor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1364,7 +1363,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("xor result") == Field::zero());
+                        assert!(cs.get("xor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
 
@@ -1732,7 +1731,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and result") == Field::one());
+                        assert!(cs.get("and result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1740,7 +1739,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and result") == Field::zero());
+                        assert!(cs.get("and result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1748,7 +1747,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1756,7 +1755,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::one());
+                        assert!(cs.get("and not result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
 
@@ -1768,7 +1767,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and result") == Field::zero());
+                        assert!(cs.get("and result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1776,7 +1775,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and result") == Field::zero());
+                        assert!(cs.get("and result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1784,7 +1783,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1792,7 +1791,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
 
@@ -1807,7 +1806,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1815,7 +1814,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1823,7 +1822,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("nor result") == Field::zero());
+                        assert!(cs.get("nor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1831,7 +1830,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("nor result") == Field::zero());
+                        assert!(cs.get("nor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
 
@@ -1846,7 +1845,7 @@ mod test {
                         OperandType::AllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::one());
+                        assert!(cs.get("and not result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
                     (
@@ -1854,7 +1853,7 @@ mod test {
                         OperandType::AllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("and not result") == Field::zero());
+                        assert!(cs.get("and not result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1862,7 +1861,7 @@ mod test {
                         OperandType::NegatedAllocatedTrue,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("nor result") == Field::zero());
+                        assert!(cs.get("nor result") == Fr::zero());
                         assert_eq!(v.value, Some(false));
                     },
                     (
@@ -1870,7 +1869,7 @@ mod test {
                         OperandType::NegatedAllocatedFalse,
                         Boolean::Is(ref v),
                     ) => {
-                        assert!(cs.get("nor result") == Field::one());
+                        assert!(cs.get("nor result") == Fr::one());
                         assert_eq!(v.value, Some(true));
                     },
 
diff --git a/r1cs-std/src/bits/uint32.rs b/r1cs-std/src/bits/uint32.rs
index 8349df67a..eb42a32f1 100644
--- a/r1cs-std/src/bits/uint32.rs
+++ b/r1cs-std/src/bits/uint32.rs
@@ -344,7 +344,8 @@ impl<ConstraintF: Field> ConditionalEqGadget<ConstraintF> for UInt32 {
 mod test {
     use super::UInt32;
     use crate::{bits::boolean::Boolean, test_constraint_system::TestConstraintSystem};
-    use algebra::fields::{bls12_381::Fr, Field};
+    use algebra::fields::bls12_381::Fr;
+    use num_traits::{One, Zero};
     use r1cs_core::ConstraintSystem;
     use rand::{Rng, SeedableRng};
     use rand_xorshift::XorShiftRng;
@@ -500,9 +501,9 @@ mod test {
 
             // Flip a bit_gadget and see if the addition constraint still works
             if cs.get("addition/result bit_gadget 0/boolean").is_zero() {
-                cs.set("addition/result bit_gadget 0/boolean", Field::one());
+                cs.set("addition/result bit_gadget 0/boolean", Fr::one());
             } else {
-                cs.set("addition/result bit_gadget 0/boolean", Field::zero());
+                cs.set("addition/result bit_gadget 0/boolean", Fr::zero());
             }
 
             assert!(!cs.is_satisfied());
diff --git a/r1cs-std/src/fields/fp12.rs b/r1cs-std/src/fields/fp12.rs
index 22f5be856..1ae353abf 100644
--- a/r1cs-std/src/fields/fp12.rs
+++ b/r1cs-std/src/fields/fp12.rs
@@ -8,6 +8,7 @@ use algebra::{
     },
     BitIterator, Field, PrimeField,
 };
+use num_traits::One;
 use std::{borrow::Borrow, marker::PhantomData};
 
 use crate::{prelude::*, Assignment};
diff --git a/r1cs-std/src/groups/curves/short_weierstrass/bls12/mod.rs b/r1cs-std/src/groups/curves/short_weierstrass/bls12/mod.rs
index 88f333aec..7c5ee8b7e 100644
--- a/r1cs-std/src/groups/curves/short_weierstrass/bls12/mod.rs
+++ b/r1cs-std/src/groups/curves/short_weierstrass/bls12/mod.rs
@@ -3,6 +3,7 @@ use algebra::{
     fields::Field,
     BitIterator, ProjectiveCurve,
 };
+use num_traits::One;
 use r1cs_core::{ConstraintSystem, SynthesisError};
 
 use crate::{
diff --git a/r1cs-std/src/groups/curves/short_weierstrass/mod.rs b/r1cs-std/src/groups/curves/short_weierstrass/mod.rs
index 55ad019b3..77801b021 100644
--- a/r1cs-std/src/groups/curves/short_weierstrass/mod.rs
+++ b/r1cs-std/src/groups/curves/short_weierstrass/mod.rs
@@ -5,6 +5,7 @@ use algebra::{
     },
     AffineCurve, BitIterator, Field, PrimeField, ProjectiveCurve,
 };
+use num_traits::{One, Zero};
 use r1cs_core::{ConstraintSystem, SynthesisError};
 use std::{borrow::Borrow, marker::PhantomData, ops::Neg};
 
diff --git a/r1cs-std/src/groups/curves/twisted_edwards/mod.rs b/r1cs-std/src/groups/curves/twisted_edwards/mod.rs
index 4951131cd..720a8d964 100644
--- a/r1cs-std/src/groups/curves/twisted_edwards/mod.rs
+++ b/r1cs-std/src/groups/curves/twisted_edwards/mod.rs
@@ -5,6 +5,7 @@ use algebra::{
     },
     BitIterator, Field,
 };
+use num_traits::{One, Zero};
 
 use r1cs_core::{ConstraintSystem, SynthesisError};
 
@@ -38,7 +39,7 @@ pub struct MontgomeryAffineGadget<
 mod montgomery_affine_impl {
     use super::*;
     use crate::Assignment;
-    use algebra::{twisted_edwards_extended::GroupAffine, AffineCurve, Field};
+    use algebra::{twisted_edwards_extended::GroupAffine, Field};
     use std::ops::{AddAssign, MulAssign, SubAssign};
 
     impl<P: TEModelParameters, ConstraintF: Field, F: FieldGadget<P::BaseField, ConstraintF>>
diff --git a/r1cs-std/src/pairing/mod.rs b/r1cs-std/src/pairing/mod.rs
index e1ad84cdc..353dab3fc 100644
--- a/r1cs-std/src/pairing/mod.rs
+++ b/r1cs-std/src/pairing/mod.rs
@@ -60,6 +60,7 @@ mod test {
     // use rand;
     use crate::test_constraint_system::TestConstraintSystem;
     use algebra::{BitIterator, Field};
+    use num_traits::One;
     use r1cs_core::ConstraintSystem;
 
     #[test]