Remove SquareRootField, and move functionality to Field

ec/src/hashing/curve_maps/swu/mod.rs

@@ -1,5 +1,5 @@
 use crate::models::short_weierstrass::SWCurveConfig;
-use ark_ff::{BigInteger, Field, One, PrimeField, SquareRootField, Zero};
+use ark_ff::{BigInteger, Field, One, PrimeField, Zero};
 use ark_std::string::ToString;
 use core::marker::PhantomData;
 

ec/src/hashing/tests/mod.rs

@@ -12,11 +12,10 @@ use crate::{
     CurveConfig,
 };
 use ark_ff::{
-    biginteger::BigInteger64, field_hashers::DefaultFieldHasher, fields::Fp64, BigInt, MontBackend,
-    MontFp,
+    biginteger::BigInteger64, field_hashers::DefaultFieldHasher, fields::Fp64, BigInt, Field,
+    MontBackend, MontFp,
 };
 
-use ark_ff::SquareRootField;
 use ark_std::vec::Vec;
 use ark_test_curves::bls12_381::{Fq, Fq2, Fq6};
 use hashbrown::HashMap;
@@ -119,7 +118,7 @@ fn test_field_division() {
 /// Check that the hashing parameters are sane: zeta should be a non-square
 #[test]
 fn checking_the_hashing_parameters() {
-    assert!(!SquareRootField::legendre(&TestSWUMapToCurveParams::ZETA).is_qr());
+    assert!(!Field::legendre(&TestSWUMapToCurveParams::ZETA).is_qr());
 }
 
 /// The point of the test is to get a simple SWU compatible curve and make

ec/src/lib.rs

@@ -20,7 +20,7 @@ extern crate derivative;
 extern crate ark_std;
 
 use ark_ff::{
-    fields::{Field, PrimeField, SquareRootField},
+    fields::{Field, PrimeField},
     UniformRand,
 };
 use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
@@ -50,7 +50,7 @@ pub mod wnaf;
 /// how to compute a pairing over a pairing-friendly curve.
 pub trait PairingEngine: Sized + 'static + Copy + Debug + Sync + Send + Eq + PartialEq {
     /// This is the scalar field of the G1/G2 groups.
-    type Fr: PrimeField + SquareRootField;
+    type Fr: PrimeField;
 
     /// The projective representation of an element in G1.
     type G1Projective: ProjectiveCurve<BaseField = Self::Fq, ScalarField = Self::Fr, Affine = Self::G1Affine>
@@ -87,10 +87,10 @@ pub trait PairingEngine: Sized + 'static + Copy + Debug + Sync + Send + Eq + Par
     type G2Prepared: Default + Clone + Send + Sync + Debug + From<Self::G2Affine>;
 
     /// The base field that hosts G1.
-    type Fq: PrimeField + SquareRootField;
+    type Fq: PrimeField;
 
     /// The extension field that hosts G2.
-    type Fqe: SquareRootField;
+    type Fqe: Field;
 
     /// The extension field that hosts the target group of the pairing.
     type Fqk: Field;
@@ -161,7 +161,7 @@ pub trait ProjectiveCurve:
     + From<<Self as ProjectiveCurve>::Affine>
 {
     type Config: CurveConfig<ScalarField = Self::ScalarField, BaseField = Self::BaseField>;
-    type ScalarField: PrimeField + SquareRootField;
+    type ScalarField: PrimeField;
     type BaseField: Field;
     type Affine: AffineCurve<
             Config = Self::Config,
@@ -251,7 +251,7 @@ pub trait AffineCurve:
 
     /// The group defined by this curve has order `h * r` where `r` is a large
     /// prime. `Self::ScalarField` is the prime field defined by `r`
-    type ScalarField: PrimeField + SquareRootField + Into<<Self::ScalarField as PrimeField>::BigInt>;
+    type ScalarField: PrimeField + Into<<Self::ScalarField as PrimeField>::BigInt>;
 
     /// The finite field over which this curve is defined.
     type BaseField: Field;

ec/src/models/bls12/mod.rs

@@ -6,7 +6,7 @@ use ark_ff::fields::{
     fp12_2over3over2::{Fp12, Fp12Config},
     fp2::Fp2Config,
     fp6_3over2::Fp6Config,
-    BitIteratorBE, Field, Fp2, PrimeField, SquareRootField,
+    BitIteratorBE, Field, Fp2, PrimeField,
 };
 use core::marker::PhantomData;
 use num_traits::{One, Zero};
@@ -33,7 +33,7 @@ pub trait Bls12Parameters: 'static {
     /// What kind of twist is this?
     const TWIST_TYPE: TwistType;
 
-    type Fp: PrimeField + SquareRootField + Into<<Self::Fp as PrimeField>::BigInt>;
+    type Fp: PrimeField + Into<<Self::Fp as PrimeField>::BigInt>;
     type Fp2Config: Fp2Config<Fp = Self::Fp>;
     type Fp6Config: Fp6Config<Fp2Config = Self::Fp2Config>;
     type Fp12Config: Fp12Config<Fp6Config = Self::Fp6Config>;

ec/src/models/bn/mod.rs

@@ -6,7 +6,7 @@ use ark_ff::fields::{
     fp12_2over3over2::{Fp12, Fp12Config},
     fp2::Fp2Config,
     fp6_3over2::Fp6Config,
-    Field, Fp2, PrimeField, SquareRootField,
+    Field, Fp2, PrimeField,
 };
 use num_traits::One;
 
@@ -31,7 +31,7 @@ pub trait BnParameters: 'static {
     const TWIST_TYPE: TwistType;
     const TWIST_MUL_BY_Q_X: Fp2<Self::Fp2Config>;
     const TWIST_MUL_BY_Q_Y: Fp2<Self::Fp2Config>;
-    type Fp: PrimeField + SquareRootField + Into<<Self::Fp as PrimeField>::BigInt>;
+    type Fp: PrimeField + Into<<Self::Fp as PrimeField>::BigInt>;
     type Fp2Config: Fp2Config<Fp = Self::Fp>;
     type Fp6Config: Fp6Config<Fp2Config = Self::Fp2Config>;
     type Fp12Config: Fp12Config<Fp6Config = Self::Fp6Config>;

ec/src/models/bw6/mod.rs

@@ -5,7 +5,7 @@ use crate::{
 use ark_ff::fields::{
     fp3::Fp3Config,
     fp6_2over3::{Fp6, Fp6Config},
-    BitIteratorBE, Field, PrimeField, SquareRootField,
+    BitIteratorBE, Field, PrimeField,
 };
 use num_traits::One;
 
@@ -24,7 +24,7 @@ pub trait BW6Parameters: 'static + Eq + PartialEq {
     const ATE_LOOP_COUNT_2: &'static [i8];
     const ATE_LOOP_COUNT_2_IS_NEGATIVE: bool;
     const TWIST_TYPE: TwistType;
-    type Fp: PrimeField + SquareRootField + Into<<Self::Fp as PrimeField>::BigInt>;
+    type Fp: PrimeField + Into<<Self::Fp as PrimeField>::BigInt>;
     type Fp3Config: Fp3Config<Fp = Self::Fp>;
     type Fp6Config: Fp6Config<Fp3Config = Self::Fp3Config>;
     type G1Parameters: SWCurveConfig<BaseField = Self::Fp>;

ec/src/models/mnt4/mod.rs

@@ -5,7 +5,7 @@ use crate::{
 use ark_ff::{
     fp2::{Fp2, Fp2Config},
     fp4::{Fp4, Fp4Config},
-    BitIteratorBE, Field, PrimeField, SquareRootField,
+    BitIteratorBE, Field, PrimeField,
 };
 use num_traits::{One, Zero};
 
@@ -30,8 +30,8 @@ pub trait MNT4Parameters: 'static {
     const FINAL_EXPONENT_LAST_CHUNK_1: <Self::Fp as PrimeField>::BigInt;
     const FINAL_EXPONENT_LAST_CHUNK_W0_IS_NEG: bool;
     const FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0: <Self::Fp as PrimeField>::BigInt;
-    type Fp: PrimeField + SquareRootField + Into<<Self::Fp as PrimeField>::BigInt>;
-    type Fr: PrimeField + SquareRootField + Into<<Self::Fr as PrimeField>::BigInt>;
+    type Fp: PrimeField + Into<<Self::Fp as PrimeField>::BigInt>;
+    type Fr: PrimeField + Into<<Self::Fr as PrimeField>::BigInt>;
     type Fp2Config: Fp2Config<Fp = Self::Fp>;
     type Fp4Config: Fp4Config<Fp2Config = Self::Fp2Config>;
     type G1Parameters: SWCurveConfig<BaseField = Self::Fp, ScalarField = Self::Fr>;

ec/src/models/mnt6/mod.rs

@@ -5,7 +5,7 @@ use crate::{
 use ark_ff::{
     fp3::{Fp3, Fp3Config},
     fp6_2over3::{Fp6, Fp6Config},
-    BitIteratorBE, Field, PrimeField, SquareRootField,
+    BitIteratorBE, Field, PrimeField,
 };
 use num_traits::{One, Zero};
 
@@ -30,8 +30,8 @@ pub trait MNT6Parameters: 'static {
     const FINAL_EXPONENT_LAST_CHUNK_1: <Self::Fp as PrimeField>::BigInt;
     const FINAL_EXPONENT_LAST_CHUNK_W0_IS_NEG: bool;
     const FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0: <Self::Fp as PrimeField>::BigInt;
-    type Fp: PrimeField + SquareRootField + Into<<Self::Fp as PrimeField>::BigInt>;
-    type Fr: PrimeField + SquareRootField + Into<<Self::Fr as PrimeField>::BigInt>;
+    type Fp: PrimeField + Into<<Self::Fp as PrimeField>::BigInt>;
+    type Fr: PrimeField + Into<<Self::Fr as PrimeField>::BigInt>;
     type Fp3Config: Fp3Config<Fp = Self::Fp>;
     type Fp6Config: Fp6Config<Fp3Config = Self::Fp3Config>;
     type G1Parameters: SWCurveConfig<BaseField = Self::Fp, ScalarField = Self::Fr>;

ec/src/models/mod.rs

@@ -1,4 +1,4 @@
-use ark_ff::{Field, PrimeField, SquareRootField};
+use ark_ff::{Field, PrimeField};
 
 pub mod bls12;
 pub mod bn;
@@ -16,10 +16,10 @@ pub mod twisted_edwards;
 /// prime-order subgroup of the curve.
 pub trait CurveConfig: Send + Sync + Sized + 'static {
     /// Base field that the curve is defined over.
-    type BaseField: Field + SquareRootField;
+    type BaseField: Field;
     /// Finite prime field corresponding to an appropriate prime-order subgroup
     /// of the curve group.
-    type ScalarField: PrimeField + SquareRootField + Into<<Self::ScalarField as PrimeField>::BigInt>;
+    type ScalarField: PrimeField + Into<<Self::ScalarField as PrimeField>::BigInt>;
 
     const COFACTOR: &'static [u64];
     const COFACTOR_INV: Self::ScalarField;

ec/src/models/short_weierstrass.rs

@@ -11,7 +11,7 @@ use ark_std::{
 };
 
 use ark_ff::{
-    fields::{Field, PrimeField, SquareRootField},
+    fields::{Field, PrimeField},
     ToConstraintField, UniformRand,
 };
 

ec/src/models/twisted_edwards.rs

@@ -18,7 +18,7 @@ use num_traits::{One, Zero};
 use zeroize::Zeroize;
 
 use ark_ff::{
-    fields::{Field, PrimeField, SquareRootField},
+    fields::{Field, PrimeField},
     ToConstraintField, UniformRand,
 };
 

ff/src/fields/arithmetic.rs

@@ -1,5 +1,5 @@
 macro_rules! sqrt_impl {
-    ($Self:ident, $P:tt, $self:expr) => {{
+    ($Self:ident, $self:expr, $two_adicity:expr, $trace_minus_one_div_two:expr, $quadratic_nonresidue:expr) => {{
         // https://eprint.iacr.org/2012/685.pdf (page 12, algorithm 5)
         // Actually this is just normal Tonelli-Shanks; since `P::Generator`
         // is a quadratic non-residue, `P::ROOT_OF_UNITY = P::GENERATOR ^ t`
@@ -10,12 +10,13 @@ macro_rules! sqrt_impl {
         // Try computing the square root (x at the end of the algorithm)
         // Check at the end of the algorithm if x was a square root
         // Begin Tonelli-Shanks
-        let mut z = $Self::qnr_to_t();
-        let mut w = $self.pow($P::TRACE_MINUS_ONE_DIV_TWO);
+        let mut z = *$quadratic_nonresidue;
+        let mut w = $self.pow($trace_minus_one_div_two);
         let mut x = w * $self;
         let mut b = x * &w;
 
-        let mut v = $P::TWO_ADICITY as usize;
+        // let mut v = $P::TWO_ADICITY as usize;
+        let mut v = *$two_adicity as usize;
 
         while !b.is_one() {
             let mut k = 0usize;
@@ -27,7 +28,7 @@ macro_rules! sqrt_impl {
                 k += 1;
             }
 
-            if k == ($P::TWO_ADICITY as usize) {
+            if k == (*$two_adicity as usize) {
                 // We are in the case where self^(T * 2^k) = x^(P::MODULUS - 1) = 1,
                 // which means that no square root exists.
                 return None;

ff/src/fields/mod.rs

@@ -130,6 +130,9 @@ pub trait Field:
     + From<bool>
 {
     type BasePrimeField: PrimeField;
+
+    const SQRT_PRECOMP: Option<SqrtPrecomputation<Self>>;
+
     type BasePrimeFieldIter: Iterator<Item = Self::BasePrimeField>;
 
     /// The additive identity of the field.
@@ -177,6 +180,19 @@ pub trait Field:
     /// from a hash-function or RNG output.
     fn from_random_bytes_with_flags<F: Flags>(bytes: &[u8]) -> Option<(Self, F)>;
 
+    /// Returns a `LegendreSymbol`, which indicates whether this field element
+    /// is  1 : a quadratic residue
+    ///  0 : equal to 0
+    /// -1 : a quadratic non-residue
+    fn legendre(&self) -> LegendreSymbol;
+
+    /// Returns the square root of self, if it exists.
+    #[must_use]
+    fn sqrt(&self) -> Option<Self>;
+
+    /// Sets `self` to be the square root of `self`, if it exists.
+    fn sqrt_in_place(&mut self) -> Option<&mut Self>;
+
     /// Returns `self * self`.
     #[must_use]
     fn square(&self) -> Self;
@@ -407,30 +423,13 @@ pub trait PrimeField:
     }
 }
 
-/// The interface for a field that supports an efficient square-root operation.
-pub trait SquareRootField: Field {
-    /// Returns a `LegendreSymbol`, which indicates whether this field element
-    /// is  
-    /// - 1: a quadratic residue
-    /// - 0: equal to 0
-    /// - -1: a quadratic non-residue
-    fn legendre(&self) -> LegendreSymbol;
-
-    /// Returns the square root of self, if it exists.
-    #[must_use]
-    fn sqrt(&self) -> Option<Self>;
-
-    /// Sets `self` to be the square root of `self`, if it exists.
-    fn sqrt_in_place(&mut self) -> Option<&mut Self>;
-}
-
 /// Indication of the field element's quadratic residuosity
 ///
 /// # Examples
 /// ```
 /// # use ark_std::test_rng;
 /// # use ark_std::UniformRand;
-/// # use ark_test_curves::{LegendreSymbol, Field, SquareRootField, bls12_381::Fq as Fp};
+/// # use ark_test_curves::{LegendreSymbol, Field, bls12_381::Fq as Fp};
 /// let a: Fp = Fp::rand(&mut test_rng());
 /// let b = a.square();
 /// assert_eq!(b.legendre(), LegendreSymbol::QuadraticResidue);
@@ -449,7 +448,7 @@ impl LegendreSymbol {
     /// ```
     /// # use ark_std::test_rng;
     /// # use ark_std::UniformRand;
-    /// # use ark_test_curves::{LegendreSymbol, Field, SquareRootField, bls12_381::Fq as Fp};
+    /// # use ark_test_curves::{LegendreSymbol, Field, bls12_381::Fq as Fp};
     /// let a: Fp = Fp::rand(&mut test_rng());
     /// let b: Fp = a.square();
     /// assert!(!b.legendre().is_zero());
@@ -462,7 +461,7 @@ impl LegendreSymbol {
     ///
     /// # Examples
     /// ```
-    /// # use ark_test_curves::{Fp2Config, LegendreSymbol, SquareRootField, bls12_381::{Fq, Fq2Config}};
+    /// # use ark_test_curves::{Fp2Config, Field, LegendreSymbol, bls12_381::{Fq, Fq2Config}};
     /// let a: Fq = Fq2Config::NONRESIDUE;
     /// assert!(a.legendre().is_qnr());
     /// ```
@@ -476,7 +475,7 @@ impl LegendreSymbol {
     /// # use ark_std::test_rng;
     /// # use ark_test_curves::bls12_381::Fq as Fp;
     /// # use ark_std::UniformRand;
-    /// # use ark_ff::{LegendreSymbol, Field, SquareRootField};
+    /// # use ark_ff::{LegendreSymbol, Field};
     /// let a: Fp = Fp::rand(&mut test_rng());
     /// let b: Fp = a.square();
     /// assert!(b.legendre().is_qr());
@@ -486,6 +485,31 @@ impl LegendreSymbol {
     }
 }
 
+pub enum SqrtPrecomputation<F: Field> {
+    // Tonelli-Shanks algorithm works for all elements, no matter what the modulus is.
+    TonelliShanks(u32, &'static [u64], F),
+}
+
+impl<F: Field> SqrtPrecomputation<F> {
+    fn sqrt(&self, elem: &F) -> Option<F> {
+        match self {
+            SqrtPrecomputation::TonelliShanks(
+                two_adicity,
+                trace_minus_one_div_two,
+                quad_non_residue_to_t,
+            ) => {
+                sqrt_impl!(
+                    F,
+                    elem,
+                    two_adicity,
+                    trace_minus_one_div_two,
+                    quad_non_residue_to_t
+                )
+            },
+        }
+    }
+}
+
 /// Iterates over a slice of `u64` in *big-endian* order.
 #[derive(Debug)]
 pub struct BitIteratorBE<Slice: AsRef<[u64]>> {