Add Legendre symbol for Fq and Fq2.

src/bls12_381/fq.rs

@@ -1,4 +1,4 @@
-use ::{Field, PrimeField, SqrtField, PrimeFieldRepr, PrimeFieldDecodingError};
+use ::{Field, LegendreField, PrimeField, SqrtField, PrimeFieldRepr, PrimeFieldDecodingError};
 use std::cmp::Ordering;
 use super::fq2::Fq2;
 
@@ -832,6 +832,23 @@ impl SqrtField for Fq {
     }
 }
 
+impl LegendreField for Fq {
+    fn legendre(&self) -> i32 {
+        // (q - 1) / 2 =
+        // 2001204777610833696708894912867952078278441409969503942666029068062015825245418932221343814564507832018947136279893
+        let x = self.pow([0xcff7fffffffd555, 0xf55ffff58a9ffffd,
+                          0x39869507b587b120, 0x23ba5c279c2895fb,
+                          0x58dd3db21a5d66bb, 0xd0088f51cbff34d2]);
+        if x == Self::one() {
+            1
+        } else if x == Self::zero() {
+            0
+        } else {
+            -1
+        }
+    }
+}
+
 #[test]
 fn test_b_coeff() {
     assert_eq!(Fq::from_repr(FqRepr::from(4)).unwrap(), B_COEFF);
@@ -1779,3 +1796,29 @@ fn test_fq_ordering() {
 fn fq_repr_tests() {
     ::tests::repr::random_repr_tests::<FqRepr>();
 }
+#[test]
+fn test_fq_legendre() {
+    assert_eq!(1, Fq::one().legendre());
+    assert_eq!(0, Fq::zero().legendre());
+
+    let e = Fq(FqRepr([0x914577fdcc41112, 0x1a6c20f3392c28e2, 0xd53f75da0c40fd21,
+                       0xb747c10d13caf0d0, 0x0de1adc19c24d8d2, 0x2103f924191033d2]));
+    assert_eq!(-1, e.legendre());
+
+
+    // f
+    let e = Fq(FqRepr([0xe51d5292ae8126f, 0x382d60874f48db82, 0xb0dde25abc614254,
+                       0x34f4456bd18813df, 0x2c668010247ee04c, 0x44cb8bbdd7c4f1b0]));
+    assert_eq!(-1, e.legendre());
+
+    // f ** 9
+    let e = Fq(FqRepr([0x69fc8eb1b590c712, 0xd73f4fb6fd34042e, 0xb5677ef2ed0eede7,
+                       0x367d831c592848c8, 0xb60615cc44e533f5, 0x127da624461b200e]));
+
+    assert_eq!(-1, e.legendre());
+
+    let e = Fq(FqRepr([0x83c7ad9e29b7facc, 0x97b3c8fbdb50cc39, 0x9e2ccd0eb5db5e72,
+                       0xc74a00d90e1b247d, 0x90e38ef46c8d7eb7, 0x16882d6aa70bb469]));
+    assert_eq!(1, e.legendre());
+
+}

src/bls12_381/fq2.rs

@@ -1,5 +1,5 @@
 use rand::{Rng, Rand};
-use ::{Field, SqrtField};
+use ::{Field, LegendreField, SqrtField};
 use super::fq::{Fq, FROBENIUS_COEFF_FQ2_C1, NEGATIVE_ONE};
 
 use std::cmp::Ordering;
@@ -44,6 +44,17 @@ impl Fq2 {
         self.c0.sub_assign(&self.c1);
         self.c1.add_assign(&t0);
     }
+
+    /// Norm of Fq2 as extension field in i over Fq
+    pub fn norm(&self) -> Fq {
+        let mut t0 = self.c0;
+        let mut t1 = self.c1;
+        t0.square();
+        t1.square();
+        t1.add_assign(&t0);
+
+        t1
+    }
 }
 
 impl Rand for Fq2 {
@@ -185,6 +196,12 @@ impl SqrtField for Fq2 {
     }
 }
 
+impl LegendreField for Fq2 {
+    fn legendre(&self) -> i32 {
+        Fq2::norm(&self).legendre()
+    }
+}
+
 #[test]
 fn test_fq2_ordering() {
     let mut a = Fq2 {
@@ -412,6 +429,17 @@ fn test_fq2_sqrt() {
     );
 }
 
+#[test]
+fn test_fq2_legendre() {
+    // i^2 = -1
+    let mut m1 = Fq2::one();
+    m1.negate();
+    assert_eq!(1, m1.legendre());
+
+    m1.mul_by_nonresidue();
+    assert_eq!(-1, m1.legendre());
+}
+
 #[cfg(test)]
 use rand::{SeedableRng, XorShiftRng};
 

src/bls12_381/fr.rs

@@ -1,4 +1,4 @@
-use ::{Field, PrimeField, SqrtField, PrimeFieldRepr, PrimeFieldDecodingError};
+use ::{Field, LegendreField, PrimeField, SqrtField, PrimeFieldRepr, PrimeFieldDecodingError};
 
 // r = 52435875175126190479447740508185965837690552500527637822603658699938581184513
 const MODULUS: FrRepr = FrRepr([0xffffffff00000001, 0x53bda402fffe5bfe, 0x3339d80809a1d805, 0x73eda753299d7d48]);
@@ -559,8 +559,7 @@ impl SqrtField for Fr {
             return Some(*self);
         }
 
-        // if self^((r - 1) // 2) != 1
-        if self.pow([0x7fffffff80000000, 0xa9ded2017fff2dff, 0x199cec0404d0ec02, 0x39f6d3a994cebea4]) != Self::one() {
+        if self.legendre() != 1 {
             None
         } else {
             let mut c = Fr(ROOT_OF_UNITY);
@@ -598,6 +597,19 @@ impl SqrtField for Fr {
     }
 }
 
+impl LegendreField for Fr {
+        fn legendre(&self) -> i32 {
+        // (q - 1) / 2 = 26217937587563095239723870254092982918845276250263818911301829349969290592256
+        let x = self.pow([0x7fffffff80000000, 0xa9ded2017fff2dff, 0x199cec0404d0ec02, 0x39f6d3a994cebea4]);
+        if x == Self::one() {
+            1
+        } else if x == Self::zero() {
+            0
+        } else {
+            -1
+        }
+    }
+}
 #[cfg(test)]
 use rand::{SeedableRng, XorShiftRng, Rand};
 
@@ -778,6 +790,17 @@ fn test_fr_repr_sub_noborrow() {
     assert!(!x.sub_noborrow(&FrRepr([0xffffffff00000001, 0x53bda402fffe5bfe, 0x3339d80809a1d805, 0x73eda753299d7d48])))
 }
 
+#[test]
+fn test_fr_legendre() {
+    assert_eq!(1, Fr::one().legendre());
+    assert_eq!(0, Fr::zero().legendre());
+
+    let e = Fr(FrRepr([0x0dbc5349cd5664da, 0x8ac5b6296e3ae29d, 0x127cb819feceaa3b, 0x3a6b21fb03867191]));
+    assert_eq!(1, e.legendre());
+    let e = Fr(FrRepr([0x96341aefd047c045, 0x9b5f4254500a4d65, 0x1ee08223b68ac240, 0x31d9cd545c0ec7c6]));
+    assert_eq!(-1, e.legendre());
+}
+
 #[test]
 fn test_fr_repr_add_nocarry() {
     let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);

src/lib.rs

@@ -332,6 +332,14 @@ pub trait SqrtField: Field
     fn sqrt(&self) -> Option<Self>;
 }
 
+/// This trait represents an element of a field that has a Legendre symbol described for it.
+pub trait LegendreField: Field
+{
+    /// Returns the legendre symbol of the field element.
+    fn legendre(&self) -> i32;
+}
+
+
 /// This trait represents a wrapper around a biginteger which can encode any element of a particular
 /// prime field. It is a smart wrapper around a sequence of `u64` limbs, least-significant digit
 /// first.