Add AdditiveGroup trait + Field: AdditiveGroup (#577)

* Add AdditiveGroup trait + Field: AdditiveGroup. This trait now integrates ZERO, double, double_in_place, and negate. Fix doctests accordingly (#507)
* Fix disambiguation of what is a ScalarField in bench-templates. Remove `ScalarField` from `CurveGroup`

---------

Co-authored-by: Pratyush Mishra <[email protected]>

CHANGELOG.md

@@ -4,6 +4,7 @@
 
 ### Breaking changes
 
+- [\#577](https://github.com/arkworks-rs/algebra/pull/577) (`ark-ff`, `ark-ec`) Add `AdditiveGroup`, a trait for additive groups (equipped with scalar field).
 - [\#593](https://github.com/arkworks-rs/algebra/pull/593) (`ark-ec`) Change `AffineRepr::xy()` to return owned values.
 
 ### Features

bench-templates/src/macros/ec.rs

@@ -3,10 +3,10 @@ macro_rules! ec_bench {
     ($curve_name:expr, $Group:ident) => {
         $crate::paste! {
             mod [<$Group:lower>] {
-                use ark_ec::Group;
+                use ark_ec::PrimeGroup;
                 use super::*;
 
-                type Scalar = <$Group as Group>::ScalarField;
+                type Scalar = <$Group as PrimeGroup>::ScalarField;
                 fn rand(c: &mut $crate::criterion::Criterion) {
                     let name = format!("{}::{}", $curve_name, stringify!($Group));
                     use ark_std::UniformRand;
@@ -18,11 +18,12 @@ macro_rules! ec_bench {
                 }
 
                 fn arithmetic(c: &mut $crate::criterion::Criterion) {
-                    use ark_ec::{CurveGroup, Group};
+                    use ark_ff::AdditiveGroup;
+                    use ark_ec::{CurveGroup, PrimeGroup};
                     use ark_std::UniformRand;
                     let name = format!("{}::{}", $curve_name, stringify!($Group));
 
-                    type Scalar = <$Group as Group>::ScalarField;
+                    type Scalar = <$Group as PrimeGroup>::ScalarField;
                     const SAMPLES: usize = 1000;
                     let mut rng = ark_std::test_rng();
                     let mut arithmetic =

bench-templates/src/macros/field.rs

@@ -63,6 +63,8 @@ macro_rules! f_bench {
 macro_rules! field_common {
     ($bench_group_name:expr, $F:ident) => {
         fn arithmetic(c: &mut $crate::criterion::Criterion) {
+            use ark_ff::AdditiveGroup;
+
             let name = format!("{}::{}", $bench_group_name, stringify!($F));
             const SAMPLES: usize = 1000;
             let mut rng = ark_std::test_rng();

ec/README.md

@@ -13,10 +13,10 @@ Implementations of particular curves using these curve models can be found in [`
 
 ### The `Group` trait
 
-Many cryptographic protocols use as core building-blocks prime-order groups. The [`Group`](https://github.com/arkworks-rs/algebra/blob/master/ec/src/lib.rs) trait is an abstraction that represents elements of such abelian prime-order groups. It provides methods for performing common operations on group elements:
+Many cryptographic protocols use as core building-blocks prime-order groups. The [`PrimeGroup`](https://github.com/arkworks-rs/algebra/blob/master/ec/src/lib.rs) trait is an abstraction that represents elements of such abelian prime-order groups. It provides methods for performing common operations on group elements:
 
 ```rust
-use ark_ec::Group;
+use ark_ec::{AdditiveGroup, PrimeGroup};
 use ark_ff::{PrimeField, Field};
 // We'll use the BLS12-381 G1 curve for this example.
 // This group has a prime order `r`, and is associated with a prime field `Fr`.
@@ -49,12 +49,12 @@ assert_eq!(f, c);
 
 ## Scalar multiplication
 
-While the `Group` trait already produces scalar multiplication routines, in many cases one can take advantage of
+While the `PrimeGroup` trait already produces scalar multiplication routines, in many cases one can take advantage of
 the group structure to perform scalar multiplication more efficiently. To allow such specialization, `ark-ec` provides
 the `ScalarMul` and `VariableBaseMSM` traits. The latter trait computes an "inner product" between a vector of scalars `s` and a vector of group elements `g`. That is, it computes `s.iter().zip(g).map(|(s, g)| g * s).sum()`.
 
 ```rust
-use ark_ec::{Group, VariableBaseMSM};
+use ark_ec::{PrimeGroup, VariableBaseMSM};
 use ark_ff::{PrimeField, Field};
 // We'll use the BLS12-381 G1 curve for this example.
 // This group has a prime order `r`, and is associated with a prime field `Fr`.
@@ -72,7 +72,7 @@ let s2 = ScalarField::rand(&mut rng);
 // Note that we're using the `GAffine` type here, as opposed to `G`.
 // This is because MSMs are more efficient when the group elements are in affine form. (See below for why.)
 //
-// The `VariableBaseMSM` trait allows specializing the input group element representation to allow 
+// The `VariableBaseMSM` trait allows specializing the input group element representation to allow
 // for more efficient implementations.
 let r = G::msm(&[a, b], &[s1, s2]).unwrap();
 assert_eq!(r, a * s1 + b * s2);
@@ -90,7 +90,7 @@ but is slower for most arithmetic operations. Let's explore how and when to use
 these:
 
 ```rust
-use ark_ec::{AffineRepr, Group, CurveGroup, VariableBaseMSM};
+use ark_ec::{AdditiveGroup, AffineRepr, PrimeGroup, CurveGroup, VariableBaseMSM};
 use ark_ff::{PrimeField, Field};
 use ark_test_curves::bls12_381::{G1Projective as G, G1Affine as GAffine, Fr as ScalarField};
 use ark_std::{Zero, UniformRand};
@@ -105,9 +105,9 @@ assert_eq!(a_aff, a);
 // We can also convert back to the `CurveGroup` representation:
 assert_eq!(a, a_aff.into_group());
 
-// As a general rule, most group operations are slower when elements 
-// are represented as `AffineRepr`. However, adding an `AffineRepr` 
-// point to a `CurveGroup` one is usually slightly more efficient than 
+// As a general rule, most group operations are slower when elements
+// are represented as `AffineRepr`. However, adding an `AffineRepr`
+// point to a `CurveGroup` one is usually slightly more efficient than
 // adding two `CurveGroup` points.
 let d = a + a_aff;
 assert_eq!(d, a.double());

ec/src/lib.rs

@@ -28,13 +28,14 @@ use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
 use ark_std::{
     fmt::{Debug, Display},
     hash::Hash,
-    ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign},
+    ops::{Add, AddAssign, Mul, MulAssign},
     vec::Vec,
 };
-use num_traits::Zero;
 pub use scalar_mul::{variable_base::VariableBaseMSM, ScalarMul};
 use zeroize::Zeroize;
 
+pub use ark_ff::AdditiveGroup;
+
 pub mod models;
 pub use self::models::*;
 
@@ -47,57 +48,14 @@ pub mod hashing;
 pub mod pairing;
 
 /// Represents (elements of) a group of prime order `r`.
-pub trait Group:
-    Eq
-    + 'static
-    + Sized
-    + CanonicalSerialize
-    + CanonicalDeserialize
-    + Copy
-    + Clone
-    + Default
-    + Send
-    + Sync
-    + Hash
-    + Debug
-    + Display
-    + UniformRand
-    + Zeroize
-    + Zero
-    + Neg<Output = Self>
-    + Add<Self, Output = Self>
-    + Sub<Self, Output = Self>
-    + Mul<<Self as Group>::ScalarField, Output = Self>
-    + AddAssign<Self>
-    + SubAssign<Self>
-    + MulAssign<<Self as Group>::ScalarField>
-    + for<'a> Add<&'a Self, Output = Self>
-    + for<'a> Sub<&'a Self, Output = Self>
-    + for<'a> Mul<&'a <Self as Group>::ScalarField, Output = Self>
-    + for<'a> AddAssign<&'a Self>
-    + for<'a> SubAssign<&'a Self>
-    + for<'a> MulAssign<&'a <Self as Group>::ScalarField>
-    + core::iter::Sum<Self>
-    + for<'a> core::iter::Sum<&'a Self>
-{
+pub trait PrimeGroup: AdditiveGroup<Scalar = Self::ScalarField> {
     /// The scalar field `F_r`, where `r` is the order of this group.
     type ScalarField: PrimeField;
 
     /// Returns a fixed generator of this group.
     #[must_use]
     fn generator() -> Self;
 
-    /// Doubles `self`.
-    #[must_use]
-    fn double(&self) -> Self {
-        let mut copy = *self;
-        copy.double_in_place();
-        copy
-    }
-
-    /// Double `self` in place.
-    fn double_in_place(&mut self) -> &mut Self;
-
     /// Performs scalar multiplication of this element.
     fn mul_bigint(&self, other: impl AsRef<[u64]>) -> Self;
 
@@ -121,7 +79,7 @@ pub trait Group:
 ///
 /// The point is guaranteed to be in the correct prime order subgroup.
 pub trait CurveGroup:
-    Group
+    PrimeGroup
     + Add<Self::Affine, Output = Self>
     + AddAssign<Self::Affine>
     // + for<'a> Add<&'a Self::Affine, Output = Self>
@@ -278,7 +236,7 @@ where
     Self::E2: MulAssign<<Self::E1 as CurveGroup>::BaseField>,
 {
     type E1: CurveGroup<
-        BaseField = <Self::E2 as Group>::ScalarField,
+        BaseField = <Self::E2 as PrimeGroup>::ScalarField,
         ScalarField = <Self::E2 as CurveGroup>::BaseField,
     >;
     type E2: CurveGroup;
@@ -289,12 +247,12 @@ pub trait PairingFriendlyCycle: CurveCycle {
     type Engine1: pairing::Pairing<
         G1 = Self::E1,
         G1Affine = <Self::E1 as CurveGroup>::Affine,
-        ScalarField = <Self::E1 as Group>::ScalarField,
+        ScalarField = <Self::E1 as PrimeGroup>::ScalarField,
     >;
 
     type Engine2: pairing::Pairing<
         G1 = Self::E2,
         G1Affine = <Self::E2 as CurveGroup>::Affine,
-        ScalarField = <Self::E2 as Group>::ScalarField,
+        ScalarField = <Self::E2 as PrimeGroup>::ScalarField,
     >;
 }

ec/src/models/bls12/g2.rs

@@ -1,4 +1,4 @@
-use ark_ff::{BitIteratorBE, Field, Fp2};
+use ark_ff::{AdditiveGroup, BitIteratorBE, Field, Fp2};
 use ark_serialize::*;
 use ark_std::{vec::Vec, One};
 

ec/src/models/bn/g2.rs

@@ -1,4 +1,7 @@
-use ark_ff::fields::{Field, Fp2};
+use ark_ff::{
+    fields::{Field, Fp2},
+    AdditiveGroup,
+};
 use ark_serialize::*;
 use ark_std::vec::Vec;
 use num_traits::One;

ec/src/models/bw6/g2.rs

@@ -1,4 +1,4 @@
-use ark_ff::{BitIteratorBE, Field};
+use ark_ff::{AdditiveGroup, BitIteratorBE, Field};
 use ark_serialize::*;
 use ark_std::vec::Vec;
 use num_traits::One;

ec/src/models/mnt4/mod.rs

@@ -5,7 +5,7 @@ use crate::{
 use ark_ff::{
     fp2::{Fp2, Fp2Config},
     fp4::{Fp4, Fp4Config},
-    CyclotomicMultSubgroup, Field, PrimeField,
+    AdditiveGroup, CyclotomicMultSubgroup, Field, PrimeField,
 };
 use itertools::Itertools;
 use num_traits::{One, Zero};

ec/src/models/mnt6/g2.rs

@@ -1,5 +1,3 @@
-use core::ops::Neg;
-
 use crate::{
     mnt6::MNT6Config,
     models::mnt6::MNT6,
@@ -8,7 +6,7 @@ use crate::{
 };
 use ark_ff::fields::{Field, Fp3};
 use ark_serialize::*;
-use ark_std::vec::Vec;
+use ark_std::{ops::Neg, vec::Vec};
 use num_traits::One;
 
 pub type G2Affine<P> = Affine<<P as MNT6Config>::G2Config>;

ec/src/models/mnt6/mod.rs

@@ -5,7 +5,7 @@ use crate::{
 use ark_ff::{
     fp3::{Fp3, Fp3Config},
     fp6_2over3::{Fp6, Fp6Config},
-    CyclotomicMultSubgroup, Field, PrimeField,
+    AdditiveGroup, CyclotomicMultSubgroup, Field, PrimeField,
 };
 use itertools::Itertools;
 use num_traits::{One, Zero};

ec/src/models/short_weierstrass/affine.rs

@@ -14,7 +14,7 @@ use ark_std::{
     One, Zero,
 };
 
-use ark_ff::{fields::Field, PrimeField, ToConstraintField, UniformRand};
+use ark_ff::{fields::Field, AdditiveGroup, PrimeField, ToConstraintField, UniformRand};
 
 use zeroize::Zeroize;
 

ec/src/models/short_weierstrass/group.rs

@@ -15,7 +15,7 @@ use ark_std::{
     One, Zero,
 };
 
-use ark_ff::{fields::Field, PrimeField, ToConstraintField, UniformRand};
+use ark_ff::{fields::Field, AdditiveGroup, PrimeField, ToConstraintField, UniformRand};
 
 use zeroize::Zeroize;
 
@@ -25,7 +25,7 @@ use rayon::prelude::*;
 use super::{Affine, SWCurveConfig};
 use crate::{
     scalar_mul::{variable_base::VariableBaseMSM, ScalarMul},
-    AffineRepr, CurveGroup, Group,
+    AffineRepr, CurveGroup, PrimeGroup,
 };
 
 /// Jacobian coordinates for a point on an elliptic curve in short Weierstrass
@@ -160,13 +160,11 @@ impl<P: SWCurveConfig> Zero for Projective<P> {
     }
 }
 
-impl<P: SWCurveConfig> Group for Projective<P> {
-    type ScalarField = P::ScalarField;
+impl<P: SWCurveConfig> AdditiveGroup for Projective<P> {
+    type Scalar = P::ScalarField;
 
-    #[inline]
-    fn generator() -> Self {
-        Affine::generator().into()
-    }
+    const ZERO: Self =
+        Self::new_unchecked(P::BaseField::ONE, P::BaseField::ONE, P::BaseField::ZERO);
 
     /// Sets `self = 2 * self`. Note that Jacobian formulae are incomplete, and
     /// so doubling cannot be computed as `self + self`. Instead, this
@@ -273,6 +271,15 @@ impl<P: SWCurveConfig> Group for Projective<P> {
             self
         }
     }
+}
+
+impl<P: SWCurveConfig> PrimeGroup for Projective<P> {
+    type ScalarField = P::ScalarField;
+
+    #[inline]
+    fn generator() -> Self {
+        Affine::generator().into()
+    }
 
     #[inline]
     fn mul_bigint(&self, other: impl AsRef<[u64]>) -> Self {

ec/src/models/short_weierstrass/mod.rs

@@ -4,7 +4,7 @@ use ark_serialize::{
 };
 use ark_std::io::{Read, Write};
 
-use ark_ff::fields::Field;
+use ark_ff::{fields::Field, AdditiveGroup};
 
 use crate::{
     scalar_mul::{

ec/src/models/twisted_edwards/affine.rs

@@ -15,7 +15,7 @@ use ark_std::{
 use num_traits::{One, Zero};
 use zeroize::Zeroize;
 
-use ark_ff::{fields::Field, PrimeField, ToConstraintField, UniformRand};
+use ark_ff::{fields::Field, AdditiveGroup, PrimeField, ToConstraintField, UniformRand};
 
 use super::{Projective, TECurveConfig, TEFlags};
 use crate::AffineRepr;

ec/src/models/twisted_edwards/group.rs

@@ -15,7 +15,7 @@ use ark_std::{
     One, Zero,
 };
 
-use ark_ff::{fields::Field, PrimeField, ToConstraintField, UniformRand};
+use ark_ff::{fields::Field, AdditiveGroup, PrimeField, ToConstraintField, UniformRand};
 
 use zeroize::Zeroize;
 
@@ -25,7 +25,7 @@ use rayon::prelude::*;
 use super::{Affine, MontCurveConfig, TECurveConfig};
 use crate::{
     scalar_mul::{variable_base::VariableBaseMSM, ScalarMul},
-    AffineRepr, CurveGroup, Group,
+    AffineRepr, CurveGroup, PrimeGroup,
 };
 
 /// `Projective` implements Extended Twisted Edwards Coordinates
@@ -150,12 +150,15 @@ impl<P: TECurveConfig> Zero for Projective<P> {
     }
 }
 
-impl<P: TECurveConfig> Group for Projective<P> {
-    type ScalarField = P::ScalarField;
+impl<P: TECurveConfig> AdditiveGroup for Projective<P> {
+    type Scalar = P::ScalarField;
 
-    fn generator() -> Self {
-        Affine::generator().into()
-    }
+    const ZERO: Self = Self::new_unchecked(
+        P::BaseField::ZERO,
+        P::BaseField::ONE,
+        P::BaseField::ZERO,
+        P::BaseField::ONE,
+    );
 
     fn double_in_place(&mut self) -> &mut Self {
         // See "Twisted Edwards Curves Revisited"
@@ -190,6 +193,14 @@ impl<P: TECurveConfig> Group for Projective<P> {
 
         self
     }
+}
+
+impl<P: TECurveConfig> PrimeGroup for Projective<P> {
+    type ScalarField = P::ScalarField;
+
+    fn generator() -> Self {
+        Affine::generator().into()
+    }
 
     #[inline]
     fn mul_bigint(&self, other: impl AsRef<[u64]>) -> Self {