Multiplication + Frobenius + Inverse

src/field/crandall_field.rs

@@ -415,7 +415,7 @@ impl DivAssign for CrandallField {
 /// Reduces to a 64-bit value. The result might not be in canonical form; it could be in between the
 /// field order and `2^64`.
 #[inline]
-fn reduce128(x: u128) -> CrandallField {
+pub fn reduce128(x: u128) -> CrandallField {
     // This is Crandall's algorithm. When we have some high-order bits (i.e. with a weight of 2^64),
     // we convert them to low-order bits by multiplying by EPSILON (the logic is a simple
     // generalization of Mersenne prime reduction). The first time we do this, the product will take

src/field/extension_field.rs

@@ -1,6 +1,9 @@
-use crate::field::crandall_field::CrandallField;
+use crate::field::crandall_field::{reduce128, CrandallField};
 use crate::field::field::Field;
 use std::fmt::{Debug, Display, Formatter};
+use std::hash::{Hash, Hasher};
+use std::iter::{Product, Sum};
+use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
 pub trait QuarticFieldExtension: Field {
     type BaseField: Field;
@@ -9,19 +12,68 @@ pub trait QuarticFieldExtension: Field {
     const W: Self::BaseField;
 
     fn to_canonical_representation(&self) -> [Self::BaseField; 4];
+
+    fn is_in_basefield(&self) -> bool {
+        self.to_canonical_representation()[1..]
+            .iter()
+            .all(|x| x.is_zero())
+    }
+
+    /// Frobenius automorphisms: x -> x^p, where p is the order of BaseField.
+    fn frobenius(&self) -> Self;
+
+    fn scalar_mul(&self, c: Self::BaseField) -> Self;
 }
 
+#[derive(Copy, Clone)]
 pub struct QuarticCrandallField([CrandallField; 4]);
 
 impl QuarticFieldExtension for QuarticCrandallField {
     type BaseField = CrandallField;
     // Verifiable in Sage with
     // ``R.<x> = GF(p)[]; assert (x^4 -3).is_irreducible()`.
-    const W: Self::BaseField = CrandallField::from_canonical_u64(3);
+    const W: Self::BaseField = CrandallField(3);
 
     fn to_canonical_representation(&self) -> [Self::BaseField; 4] {
         self.0
     }
+
+    fn frobenius(&self) -> Self {
+        let [a0, a1, a2, a3] = self.to_canonical_representation();
+        let k = (Self::BaseField::ORDER - 1) / 4;
+        let z0 = Self::W.exp_usize(k as usize);
+        let mut z = Self::BaseField::ONE;
+        let b0 = a0 * z;
+        z *= z0;
+        let b1 = a1 * z;
+        z *= z0;
+        let b2 = a2 * z;
+        z *= z0;
+        let b3 = a3 * z;
+
+        Self([b0, b1, b2, b3])
+    }
+
+    fn scalar_mul(&self, c: Self::BaseField) -> Self {
+        let [a0, a1, a2, a3] = self.to_canonical_representation();
+        Self([a0 * c, a1 * c, a2 * c, a3 * c])
+    }
+}
+
+impl PartialEq for QuarticCrandallField {
+    fn eq(&self, other: &Self) -> bool {
+        self.to_canonical_representation() == other.to_canonical_representation()
+    }
+}
+
+impl Eq for QuarticCrandallField {}
+
+impl Hash for QuarticCrandallField {
+    fn hash<H: Hasher>(&self, state: &mut H) {
+        for l in &self.to_canonical_representation() {
+            Hash::hash(l, state);
+        }
+    }
 }
 
 impl Field for QuarticCrandallField {
@@ -49,7 +101,7 @@ impl Field for QuarticCrandallField {
     const ORDER: u64 = 0;
     const TWO_ADICITY: usize = 30;
     const MULTIPLICATIVE_GROUP_GENERATOR: Self = Self([
-        CrandallField::from_canonical_u64(3),
+        CrandallField(3),
         CrandallField::ONE,
         CrandallField::ZERO,
         CrandallField::ZERO,
@@ -58,11 +110,23 @@ impl Field for QuarticCrandallField {
         CrandallField::ZERO,
         CrandallField::ZERO,
         CrandallField::ZERO,
-        CrandallField::from_canonical_u64(14096607364803438105),
+        CrandallField(14096607364803438105),
     ]);
 
+    // Algorithm 11.3.4 in Handbook of Elliptic and Hyperelliptic Curve Cryptography.
     fn try_inverse(&self) -> Option<Self> {
-        todo!()
+        if self.is_zero() {
+            return None;
+        }
+
+        let a_pow_p = self.frobenius();
+        let a_pow_p_plus_1 = a_pow_p * *self;
+        let a_pow_p3_plus_p2 = a_pow_p_plus_1.frobenius().frobenius();
+        let a_pow_r_minus_1 = a_pow_p3_plus_p2 * a_pow_p;
+        let a_pow_r = a_pow_r_minus_1 * *self;
+        debug_assert!(a_pow_r.is_in_basefield());
+
+        Some(a_pow_r_minus_1.scalar_mul(a_pow_r.0[0].inverse()))
     }
 
     fn to_canonical_u64(&self) -> u64 {
@@ -89,3 +153,129 @@ impl Debug for QuarticCrandallField {
         Display::fmt(self, f)
     }
 }
+
+impl Neg for QuarticCrandallField {
+    type Output = Self;
+
+    #[inline]
+    fn neg(self) -> Self {
+        Self([-self.0[0], -self.0[1], -self.0[2], -self.0[3]])
+    }
+}
+
+impl Add for QuarticCrandallField {
+    type Output = Self;
+
+    #[inline]
+    fn add(self, rhs: Self) -> Self {
+        Self([
+            self.0[0] + rhs.0[0],
+            self.0[1] + rhs.0[1],
+            self.0[2] + rhs.0[2],
+            self.0[3] + rhs.0[3],
+        ])
+    }
+}
+
+impl AddAssign for QuarticCrandallField {
+    fn add_assign(&mut self, rhs: Self) {
+        *self = *self + rhs;
+    }
+}
+
+impl Sum for QuarticCrandallField {
+    fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
+        iter.fold(Self::ZERO, |acc, x| acc + x)
+    }
+}
+
+impl Sub for QuarticCrandallField {
+    type Output = Self;
+
+    #[inline]
+    fn sub(self, rhs: Self) -> Self {
+        Self([
+            self.0[0] - rhs.0[0],
+            self.0[1] - rhs.0[1],
+            self.0[2] - rhs.0[2],
+            self.0[3] - rhs.0[3],
+        ])
+    }
+}
+
+impl SubAssign for QuarticCrandallField {
+    #[inline]
+    fn sub_assign(&mut self, rhs: Self) {
+        *self = *self - rhs;
+    }
+}
+
+impl Mul for QuarticCrandallField {
+    type Output = Self;
+
+    #[inline]
+    fn mul(self, rhs: Self) -> Self {
+        let Self([a0, a1, a2, a3]) = self;
+        let Self([b0, b1, b2, b3]) = rhs;
+        let a0 = a0.0 as u128;
+        let a1 = a1.0 as u128;
+        let a2 = a2.0 as u128;
+        let a3 = a3.0 as u128;
+        let b0 = b0.0 as u128;
+        let b1 = b1.0 as u128;
+        let b2 = b2.0 as u128;
+        let b3 = b3.0 as u128;
+        let w = Self::W.0 as u128;
+
+        let c0 = reduce128(a0 * b0 + w * (a1 * b3 + a2 * b2 + a3 * b1));
+        let c1 = reduce128(a0 * b1 + a1 * b0 + w * (a2 * b3 + a3 * b2));
+        let c2 = reduce128(a0 * b2 + a1 * b1 + a2 * b0 + w * a3 * b3);
+        let c3 = reduce128(a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0);
+
+        Self([c0, c1, c2, c3])
+    }
+}
+
+impl MulAssign for QuarticCrandallField {
+    #[inline]
+    fn mul_assign(&mut self, rhs: Self) {
+        *self = *self * rhs;
+    }
+}
+
+impl Product for QuarticCrandallField {
+    fn product<I: Iterator<Item = Self>>(iter: I) -> Self {
+        iter.fold(Self::ONE, |acc, x| acc * x)
+    }
+}
+
+impl Div for QuarticCrandallField {
+    type Output = Self;
+
+    #[allow(clippy::suspicious_arithmetic_impl)]
+    fn div(self, rhs: Self) -> Self::Output {
+        self * rhs.inverse()
+    }
+}
+
+impl DivAssign for QuarticCrandallField {
+    fn div_assign(&mut self, rhs: Self) {
+        *self = *self / rhs;
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use crate::field::crandall_field::CrandallField;
+    use crate::field::extension_field::{QuarticCrandallField, QuarticFieldExtension};
+    use crate::field::field::Field;
+    use crate::test_arithmetic;
+
+    test_arithmetic!(crate::field::crandall_field::QuarticCrandallField);
+
+    #[test]
+    fn test_frobenius() {
+        let x = QuarticCrandallField::rand();
+        assert_eq!(x.exp_usize(CrandallField::ORDER as usize), x.frobenius());
+    }
+}