group: Remove CurveProjective::Base and CurveAffine::Base

These associated types were completly unused. The only place we need
information about the base field of an elliptic curve is inside Jubjub
when operating over its coordinates to implement EC math inside the
circuit, and we can handle that either concretely, or with a future
trait specifically for that use-case.

bellman/src/groth16/tests/dummy_engine.rs

@@ -394,7 +394,6 @@ impl PrimeGroup for Fr {}
 
 impl CurveProjective for Fr {
     type Affine = Fr;
-    type Base = Fr;
 
     fn batch_normalize(p: &[Self], q: &mut [Self::Affine]) {
         assert_eq!(p.len(), q.len());
@@ -436,7 +435,6 @@ impl CurveAffine for Fr {
     type Compressed = FakePoint;
     type Uncompressed = FakePoint;
     type Projective = Fr;
-    type Base = Fr;
     type Scalar = Fr;
 
     fn identity() -> Self {

group/src/lib.rs

@@ -1,7 +1,7 @@
 // Catch documentation errors caused by code changes.
 #![deny(intra_doc_link_resolution_failure)]
 
-use ff::{Field, PrimeField};
+use ff::PrimeField;
 use rand::RngCore;
 use std::fmt;
 use std::iter::Sum;
@@ -97,7 +97,6 @@ pub trait CurveProjective:
     + GroupOps<<Self as CurveProjective>::Affine>
     + GroupOpsOwned<<Self as CurveProjective>::Affine>
 {
-    type Base: Field;
     type Affine: CurveAffine<Projective = Self, Scalar = Self::Scalar>
         + Mul<Self::Scalar, Output = Self>
         + for<'r> Mul<Self::Scalar, Output = Self>;
@@ -136,7 +135,6 @@ pub trait CurveAffine:
     + for<'r> Mul<<Self as CurveAffine>::Scalar, Output = <Self as CurveAffine>::Projective>
 {
     type Scalar: PrimeField;
-    type Base: Field;
     type Projective: CurveProjective<Affine = Self, Scalar = Self::Scalar>;
     type Uncompressed: Default + AsRef<[u8]> + AsMut<[u8]>;
     type Compressed: Default + AsRef<[u8]> + AsMut<[u8]>;

pairing/src/bls12_381/ec.rs

@@ -200,7 +200,6 @@ macro_rules! curve_impl {
 
         impl CurveAffine for $affine {
             type Scalar = $scalarfield;
-            type Base = $basefield;
             type Projective = $projective;
             type Uncompressed = $uncompressed;
             type Compressed = $compressed;
@@ -748,7 +747,6 @@ macro_rules! curve_impl {
         impl PrimeGroup for $projective {}
 
         impl CurveProjective for $projective {
-            type Base = $basefield;
             type Affine = $affine;
 
             fn batch_normalize(p: &[Self], q: &mut [$affine]) {

pairing/src/lib.rs

@@ -30,7 +30,7 @@ use subtle::CtOption;
 /// of prime order `r`, and are equipped with a bilinear pairing function.
 pub trait Engine: ScalarEngine {
     /// The projective representation of an element in G1.
-    type G1: CurveProjective<Base = Self::Fq, Scalar = Self::Fr, Affine = Self::G1Affine>
+    type G1: CurveProjective<Scalar = Self::Fr, Affine = Self::G1Affine>
         + From<Self::G1Affine>
         + GroupOps<Self::G1Affine>
         + GroupOpsOwned<Self::G1Affine>
@@ -39,7 +39,6 @@ pub trait Engine: ScalarEngine {
 
     /// The affine representation of an element in G1.
     type G1Affine: PairingCurveAffine<
-            Base = Self::Fq,
             Scalar = Self::Fr,
             Projective = Self::G1,
             Pair = Self::G2Affine,
@@ -49,7 +48,7 @@ pub trait Engine: ScalarEngine {
         + for<'a> Mul<&'a Self::Fr, Output = Self::G1>;
 
     /// The projective representation of an element in G2.
-    type G2: CurveProjective<Base = Self::Fqe, Scalar = Self::Fr, Affine = Self::G2Affine>
+    type G2: CurveProjective<Scalar = Self::Fr, Affine = Self::G2Affine>
         + From<Self::G2Affine>
         + GroupOps<Self::G2Affine>
         + GroupOpsOwned<Self::G2Affine>
@@ -58,7 +57,6 @@ pub trait Engine: ScalarEngine {
 
     /// The affine representation of an element in G2.
     type G2Affine: PairingCurveAffine<
-            Base = Self::Fqe,
             Scalar = Self::Fr,
             Projective = Self::G2,
             Pair = Self::G1Affine,