[RFC] Convert identity functions in Field, Group, and {Projective,Aff…

…ine}Curve traits with One/Zero traits from num_traits.

- contributes to arkworks-rs#50,
- depends on arkworks-rs#53 and builds on it,
- due to coherence & requirements of `num_traits::{Zero, One}` to implement `std::ops::Add<Self, ..>` and (resp.) `std::ops::Mul<Self, ..>`, I've had to replace the afferent `impl<'a, P: ..> (Add|Mul)<&'a Self> for Group(Affine|Projective)<P>` by direct implementations on `Self`,
- I did not have to fight the borrow checker for this conversion => I think this hints arithmetic operations are called in contexts where the operand is owned,
- hence should this end up on a merge track, we may want to open an issue to convert the `impl<'a, P:..> (Neg|Sub|..)<&'a Self> for ..<P>` trait usage to direct `impl<P:..> (Neg|Sub|..)<Self> for ..<P>`
- the `impl AddAssign for GroupAffine<P>` in curves/models/short_weierstrass_jacobian.rs is provided to fit trait bounds, and without any guarantee of suitability for any particular purpose
- and that, even though I don't think it's used.

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 }

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;

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;

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]

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>;

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>;

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};
 

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;
 

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;

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]

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]);
@@ -205,26 +200,18 @@ pub trait AffineCurve:
     + Hash
     + Debug
     + Display
+    + Zero
     + Neg<Output = Self>
     + 'static
 {
     type ScalarField: PrimeField + SquareRootField + Into<<Self::ScalarField as PrimeField>::BigInt>;
     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]

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>;

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>;

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;
 

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
     }
 }