ff: Remove SqrtField trait

The sqrt() function is now part of the Field trait. ff_derive returns an
error on fields for which it does not support generating a square root
function.

Note that Fq6 and Fq12 in pairing::bls12_381 leave the function
unimplemented. They will be dropped once the migration to the bls12_381
crate is complete. The equivalent structs in that crate are not exposed.

pairing/benches/bls12_381/fq.rs

@@ -3,7 +3,7 @@ use rand_core::SeedableRng;
 use rand_xorshift::XorShiftRng;
 use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
-use ff::{Field, PrimeField, SqrtField};
+use ff::{Field, PrimeField};
 use pairing::bls12_381::*;
 
 fn bench_fq_add_assign(c: &mut Criterion) {

pairing/benches/bls12_381/fq2.rs

@@ -3,7 +3,7 @@ use rand_core::SeedableRng;
 use rand_xorshift::XorShiftRng;
 use std::ops::{AddAssign, MulAssign, SubAssign};
 
-use ff::{Field, SqrtField};
+use ff::Field;
 use pairing::bls12_381::*;
 
 fn bench_fq2_add_assign(c: &mut Criterion) {

pairing/benches/bls12_381/fr.rs

@@ -3,7 +3,7 @@ use rand_core::SeedableRng;
 use rand_xorshift::XorShiftRng;
 use std::ops::{AddAssign, MulAssign, Neg, SubAssign};
 
-use ff::{Field, PrimeField, SqrtField};
+use ff::{Field, PrimeField};
 use pairing::bls12_381::*;
 
 fn bench_fr_add_assign(c: &mut Criterion) {

pairing/src/bls12_381/ec.rs

@@ -754,7 +754,7 @@ pub mod g1 {
     use super::super::{Bls12, Fq, Fq12, FqRepr, Fr};
     use super::g2::G2Affine;
     use crate::{Engine, PairingCurveAffine};
-    use ff::{BitIterator, Field, PrimeField, SqrtField};
+    use ff::{BitIterator, Field, PrimeField};
     use group::{CurveAffine, CurveProjective, EncodedPoint, GroupDecodingError};
     use rand_core::RngCore;
     use std::fmt;
@@ -1054,8 +1054,6 @@ pub mod g1 {
 
     #[test]
     fn g1_generator() {
-        use crate::SqrtField;
-
         let mut x = Fq::zero();
         let mut i = 0;
         loop {
@@ -1366,7 +1364,7 @@ pub mod g2 {
     use super::super::{Bls12, Fq, Fq12, Fq2, FqRepr, Fr};
     use super::g1::G1Affine;
     use crate::{Engine, PairingCurveAffine};
-    use ff::{BitIterator, Field, PrimeField, SqrtField};
+    use ff::{BitIterator, Field, PrimeField};
     use group::{CurveAffine, CurveProjective, EncodedPoint, GroupDecodingError};
     use rand_core::RngCore;
     use std::fmt;
@@ -1708,8 +1706,6 @@ pub mod g2 {
 
     #[test]
     fn g2_generator() {
-        use crate::SqrtField;
-
         let mut x = Fq2::zero();
         let mut i = 0;
         loop {

pairing/src/bls12_381/fq.rs

@@ -1715,8 +1715,6 @@ fn test_fq_pow() {
 
 #[test]
 fn test_fq_sqrt() {
-    use ff::SqrtField;
-
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -1846,8 +1844,6 @@ fn test_fq_num_bits() {
 
 #[test]
 fn test_fq_root_of_unity() {
-    use ff::SqrtField;
-
     assert_eq!(Fq::S, 1);
     assert_eq!(Fq::multiplicative_generator(), Fq::from(2));
     assert_eq!(

pairing/src/bls12_381/fq12.rs

@@ -237,6 +237,10 @@ impl Field for Fq12 {
             c1: t.mul(&self.c1).neg(),
         })
     }
+
+    fn sqrt(&self) -> CtOption<Self> {
+        unimplemented!()
+    }
 }
 
 #[cfg(test)]

pairing/src/bls12_381/fq2.rs

@@ -1,5 +1,5 @@
 use super::fq::{Fq, FROBENIUS_COEFF_FQ2_C1, NEGATIVE_ONE};
-use ff::{Field, PowVartime, SqrtField};
+use ff::{Field, PowVartime};
 use rand_core::RngCore;
 use std::cmp::Ordering;
 use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
@@ -241,9 +241,7 @@ impl Field for Fq2 {
     fn frobenius_map(&mut self, power: usize) {
         self.c1.mul_assign(&FROBENIUS_COEFF_FQ2_C1[power % 2]);
     }
-}
 
-impl SqrtField for Fq2 {
     /// WARNING: THIS IS NOT ACTUALLY CONSTANT TIME YET!
     /// THIS WILL BE REPLACED BY THE bls12_381 CRATE, WHICH IS CONSTANT TIME!
     fn sqrt(&self) -> CtOption<Self> {

pairing/src/bls12_381/fq6.rs

@@ -391,6 +391,10 @@ impl Field for Fq6 {
             tmp
         })
     }
+
+    fn sqrt(&self) -> CtOption<Self> {
+        unimplemented!()
+    }
 }
 
 #[cfg(test)]

pairing/src/bls12_381/fr.rs

@@ -495,8 +495,6 @@ fn test_fr_pow() {
 
 #[test]
 fn test_fr_sqrt() {
-    use ff::SqrtField;
-
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,
@@ -628,8 +626,6 @@ fn test_fr_num_bits() {
 
 #[test]
 fn test_fr_root_of_unity() {
-    use ff::SqrtField;
-
     assert_eq!(Fr::S, 32);
     assert_eq!(Fr::multiplicative_generator(), Fr::from(7));
     assert_eq!(

pairing/src/lib.rs

@@ -20,7 +20,7 @@ pub mod tests;
 
 pub mod bls12_381;
 
-use ff::{Field, PrimeField, ScalarEngine, SqrtField};
+use ff::{Field, PrimeField, ScalarEngine};
 use group::{CurveAffine, CurveOps, CurveOpsOwned, CurveProjective};
 use subtle::CtOption;
 
@@ -61,10 +61,10 @@ pub trait Engine: ScalarEngine {
         > + From<Self::G2>;
 
     /// The base field that hosts G1.
-    type Fq: PrimeField + SqrtField;
+    type Fq: PrimeField;
 
     /// The extension field that hosts G2.
-    type Fqe: SqrtField;
+    type Fqe: Field;
 
     /// The extension field that hosts the target group of the pairing.
     type Fqk: Field;

pairing/src/tests/field.rs

@@ -1,4 +1,4 @@
-use ff::{Field, PowVartime, PrimeField, SqrtField};
+use ff::{Field, PowVartime, PrimeField};
 use rand_core::{RngCore, SeedableRng};
 use rand_xorshift::XorShiftRng;
 
@@ -23,7 +23,7 @@ pub fn random_frobenius_tests<F: Field, C: AsRef<[u8]>>(characteristic: C, maxpo
     }
 }
 
-pub fn random_sqrt_tests<F: SqrtField>() {
+pub fn random_sqrt_tests<F: Field>() {
     let mut rng = XorShiftRng::from_seed([
         0x59, 0x62, 0xbe, 0x5d, 0x76, 0x3d, 0x31, 0x8d, 0x17, 0xdb, 0x37, 0x32, 0x54, 0x06, 0xbc,
         0xe5,