Curve448 with full coordinates

ed448-goldilocks/benches/bench.rs

@@ -1,6 +1,7 @@
 use criterion::{BatchSize, Criterion, criterion_group, criterion_main};
 use ed448_goldilocks::{
-    Decaf448, DecafPoint, DecafScalar, Ed448, EdwardsPoint, EdwardsScalar, MontgomeryPoint,
+    Curve448, Decaf448, DecafPoint, DecafScalar, Ed448, EdwardsPoint, EdwardsScalar,
+    MontgomeryScalar, MontgomeryXpoint, ProjectiveMontgomeryPoint, ProjectiveMontgomeryXpoint,
 };
 use elliptic_curve::group::GroupEncoding;
 use elliptic_curve::{Field, Group};
@@ -123,22 +124,76 @@ pub fn decaf448(c: &mut Criterion) {
     group.finish();
 }
 
+pub fn curve448(c: &mut Criterion) {
+    let mut group = c.benchmark_group("Curve448");
+
+    group.bench_function("scalar multiplication", |b| {
+        b.iter_batched(
+            || {
+                let point = ProjectiveMontgomeryPoint::try_from_rng(&mut OsRng).unwrap();
+                let scalar = MontgomeryScalar::try_from_rng(&mut OsRng).unwrap();
+                (point, scalar)
+            },
+            |(point, scalar)| point * scalar,
+            BatchSize::SmallInput,
+        )
+    });
+
+    group.bench_function("point addition", |b| {
+        b.iter_batched(
+            || {
+                let p1 = ProjectiveMontgomeryPoint::try_from_rng(&mut OsRng).unwrap();
+                let p2 = ProjectiveMontgomeryPoint::try_from_rng(&mut OsRng).unwrap();
+                (p1, p2)
+            },
+            |(p1, p2)| p1 + p2,
+            BatchSize::SmallInput,
+        )
+    });
+
+    group.bench_function("encode_to_curve", |b| {
+        b.iter_batched(
+            || {
+                let mut msg = [0; 64];
+                OsRng.try_fill_bytes(&mut msg).unwrap();
+                msg
+            },
+            |msg| Curve448::encode_from_bytes(&msg, b"test DST"),
+            BatchSize::SmallInput,
+        )
+    });
+
+    group.finish();
+}
+
 pub fn x448(c: &mut Criterion) {
     let mut group = c.benchmark_group("X448");
 
     group.bench_function("scalar multiplication", |b| {
         b.iter_batched(
             || {
-                let mut point = MontgomeryPoint::default();
+                let mut point = MontgomeryXpoint::default();
                 OsRng.try_fill_bytes(&mut point.0).unwrap();
-                let scalar = EdwardsScalar::try_from_rng(&mut OsRng).unwrap();
+                let scalar = MontgomeryScalar::try_from_rng(&mut OsRng).unwrap();
                 (point, scalar)
             },
             |(point, scalar)| &point * &scalar,
             BatchSize::SmallInput,
         )
     });
 
+    group.bench_function("encode_to_curve", |b| {
+        b.iter_batched(
+            || {
+                let mut msg = [0; 64];
+                OsRng.try_fill_bytes(&mut msg).unwrap();
+                msg
+            },
+            |msg| ProjectiveMontgomeryXpoint::encode_with_defaults(&msg, b"test DST"),
+            BatchSize::SmallInput,
+        )
+    });
+
     group.finish();
 }
 

ed448-goldilocks/src/edwards/affine.rs

@@ -95,37 +95,6 @@ impl AffinePoint {
         }
     }
 
-    pub(crate) fn isogeny(&self) -> Self {
-        let x = self.x;
-        let y = self.y;
-        let mut t0 = x.square(); // x^2
-        let t1 = t0 + FieldElement::ONE; // x^2+1
-        t0 -= FieldElement::ONE; // x^2-1
-        let mut t2 = y.square(); // y^2
-        t2 = t2.double(); // 2y^2
-        let t3 = x.double(); // 2x
-
-        let mut t4 = t0 * y; // y(x^2-1)
-        t4 = t4.double(); // 2y(x^2-1)
-        let xNum = t4.double(); // xNum = 4y(x^2-1)
-
-        let mut t5 = t0.square(); // x^4-2x^2+1
-        t4 = t5 + t2; // x^4-2x^2+1+2y^2
-        let xDen = t4 + t2; // xDen = x^4-2x^2+1+4y^2
-
-        t5 *= x; // x^5-2x^3+x
-        t4 = t2 * t3; // 4xy^2
-        let yNum = t4 - t5; // yNum = -(x^5-2x^3+x-4xy^2)
-
-        t4 = t1 * t2; // 2x^2y^2+2y^2
-        let yDen = t5 - t4; // yDen = x^5-2x^3+x-2x^2y^2-2y^2
-
-        Self {
-            x: xNum * xDen.invert(),
-            y: yNum * yDen.invert(),
-        }
-    }
-
     /// Standard compression; store Y and sign of X
     pub fn compress(&self) -> CompressedEdwardsY {
         let affine_x = self.x;

ed448-goldilocks/src/edwards/extended.rs

@@ -311,8 +311,8 @@ impl EdwardsPoint {
         T: FieldElement::ZERO,
     };
 
-    /// Convert this point to [`MontgomeryPoint`]
-    pub fn to_montgomery(&self) -> MontgomeryPoint {
+    /// Convert this point to [`MontgomeryXpoint`]
+    pub fn to_montgomery_x(&self) -> MontgomeryXpoint {
         // u = y^2 * [(1-dy^2)/(1-y^2)]
 
         let affine = self.to_affine();
@@ -322,7 +322,29 @@ impl EdwardsPoint {
 
         let u = yy * (FieldElement::ONE - dyy) * (FieldElement::ONE - yy).invert();
 
-        MontgomeryPoint(u.to_bytes())
+        MontgomeryXpoint(u.to_bytes())
+    }
+
+    /// Convert this point to [`AffineMontgomeryPoint`]
+    // See https://www.rfc-editor.org/rfc/rfc7748#section-4.2 4-isogeny maps
+    pub fn to_montgomery(&self) -> AffineMontgomeryPoint {
+        // u = y^2/x^2
+        // v = (2 - x^2 - y^2)*y/x^3
+
+        let affine = self.to_affine();
+
+        // TODO: optimize to a single inversion.
+        let xx = affine.x.square();
+        let yy = affine.y.square();
+
+        let u = yy * xx.invert();
+        let v = (FieldElement::TWO - xx - yy) * affine.y * (xx * affine.x).invert();
+
+        AffineMontgomeryPoint::conditional_select(
+            &AffineMontgomeryPoint::new(u, v),
+            &AffineMontgomeryPoint::IDENTITY,
+            self.ct_eq(&Self::IDENTITY),
+        )
     }
 
     /// Generic scalar multiplication to compute s*P
@@ -787,6 +809,7 @@ mod tests {
     use proptest::prelude::any;
     use proptest::proptest;
     use rand_core::{OsRng, TryRngCore};
+    use sha3::Shake256;
 
     fn hex_to_field(hex: &'static str) -> FieldElement {
         assert_eq!(hex.len(), 56 * 2);
@@ -947,9 +970,7 @@ mod tests {
         ];
 
         for (msg, x, y) in MSGS {
-            let p =
-                hash2curve::hash_from_bytes::<Ed448, ExpandMsgXof<sha3::Shake256>>(&[msg], &[DST])
-                    .unwrap();
+            let p = Ed448::hash_from_bytes(msg, DST).unwrap();
             assert_eq!(p.is_on_curve().unwrap_u8(), 1u8);
             let p = p.to_affine();
             let mut xx = [0u8; 56];
@@ -986,11 +1007,7 @@ mod tests {
         ];
 
         for (msg, x, y) in MSGS {
-            let p = hash2curve::encode_from_bytes::<Ed448, ExpandMsgXof<sha3::Shake256>>(
-                &[msg],
-                &[DST],
-            )
-            .unwrap();
+            let p = Ed448::encode_from_bytes(msg, DST).unwrap();
             assert_eq!(p.is_on_curve().unwrap_u8(), 1u8);
             let p = p.to_affine();
             let mut xx = [0u8; 56];
@@ -1001,6 +1018,22 @@ mod tests {
             yy.reverse();
             assert_eq!(p.x.to_bytes(), xx);
             assert_eq!(p.y.to_bytes(), yy);
+
+            // Test Montgomery to Edwards conversion.
+            // See https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve/blob/664b13592116cecc9e52fb192dcde0ade36f904e/poc/ell2_opt_3mod4.sage#L243-L245.
+            let conv_p =
+                ProjectiveMontgomeryXpoint::encode::<ExpandMsgXof<Shake256>>(&[msg], &[DST])
+                    .unwrap()
+                    .to_affine();
+            let conv_p1 = conv_p.to_edwards(Choice::from(0)).unwrap();
+            let conv_p2 = conv_p.to_edwards(Choice::from(1)).unwrap();
+            assert!(conv_p1.x == p.x || conv_p2.x == p.x);
+            assert!(conv_p1.y == p.y || conv_p2.y == p.y);
+
+            let conv_p =
+                AffinePoint::from(Curve448::encode_from_bytes(msg, DST).unwrap().to_affine());
+            assert_eq!(conv_p.x, p.x);
+            assert_eq!(conv_p.y, p.y);
         }
     }
 

ed448-goldilocks/src/edwards/scalar.rs

@@ -2,8 +2,8 @@ use crate::Ed448;
 use crate::field::{CurveWithScalar, ORDER, Scalar, ScalarBytes, WideScalarBytes};
 
 use elliptic_curve::array::Array;
-use elliptic_curve::bigint::{Limb, NonZero, U448, U704};
-use elliptic_curve::consts::{U57, U84, U88};
+use elliptic_curve::bigint::{Limb, U448};
+use elliptic_curve::consts::{U57, U84};
 use elliptic_curve::ops::Reduce;
 use elliptic_curve::scalar::FromUintUnchecked;
 use subtle::{Choice, CtOption};
@@ -84,17 +84,7 @@ impl From<&EdwardsScalar> for elliptic_curve::scalar::ScalarBits<Ed448> {
 
 impl Reduce<Array<u8, U84>> for EdwardsScalar {
     fn reduce(value: &Array<u8, U84>) -> Self {
-        const SEMI_WIDE_MODULUS: NonZero<U704> = NonZero::<U704>::new_unwrap(U704::from_be_hex(
-            "00000000000000000000000000000000000000000000000000000000000000003fffffffffffffffffffffffffffffffffffffffffffffffffffffff7cca23e9c44edb49aed63690216cc2728dc58f552378c292ab5844f3",
-        ));
-        let mut tmp = Array::<u8, U88>::default();
-        tmp[4..].copy_from_slice(&value[..]);
-
-        let mut num = U704::from_be_slice(&tmp[..]);
-        num %= SEMI_WIDE_MODULUS;
-        let mut words = [0; U448::LIMBS];
-        words.copy_from_slice(&num.to_words()[..U448::LIMBS]);
-        Scalar::new(U448::from_words(words))
+        Self::from_okm_u84(value)
     }
 }
 

ed448-goldilocks/src/field/element.rs

@@ -4,8 +4,8 @@ use core::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
 use super::{ConstMontyType, MODULUS};
 use crate::{
-    AffinePoint, Decaf448, DecafPoint, Ed448, EdwardsPoint,
-    curve::twedwards::extended::ExtendedPoint as TwistedExtendedPoint,
+    AffineMontgomeryPoint, AffinePoint, Curve448, Decaf448, DecafPoint, Ed448, EdwardsPoint,
+    ProjectiveMontgomeryPoint, curve::twedwards::extended::ExtendedPoint as TwistedExtendedPoint,
 };
 use elliptic_curve::ops::Reduce;
 use elliptic_curve::{
@@ -196,7 +196,7 @@ impl MapToCurve for Ed448 {
     type Length = U84;
 
     fn map_to_curve(element: FieldElement) -> EdwardsPoint {
-        element.map_to_curve_elligator2().isogeny().to_edwards()
+        AffinePoint::from(element.map_to_curve_elligator2_curve448()).to_edwards()
     }
 }
 
@@ -272,6 +272,16 @@ impl Field for FieldElement {
     }
 }
 
+impl MapToCurve for Curve448 {
+    type SecurityLevel = U28;
+    type FieldElement = FieldElement;
+    type Length = U84;
+
+    fn map_to_curve(element: FieldElement) -> ProjectiveMontgomeryPoint {
+        element.map_to_curve_elligator2_curve448().into()
+    }
+}
+
 impl FieldElement {
     pub const A_PLUS_TWO_OVER_FOUR: Self = Self(ConstMontyType::new(&U448::from_be_hex(
         "00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000098aa",
@@ -377,6 +387,10 @@ impl FieldElement {
         Self(self.0.double())
     }
 
+    pub fn triple(&self) -> Self {
+        self.double() + self
+    }
+
     /// Computes the inverse square root of a field element
     /// Returns the result and a boolean to indicate whether self
     /// was a Quadratic residue
@@ -430,7 +444,7 @@ impl FieldElement {
         Self(self.0.div_by_2())
     }
 
-    pub(crate) fn map_to_curve_elligator2(&self) -> AffinePoint {
+    pub(crate) fn map_to_curve_elligator2_curve448(&self) -> AffineMontgomeryPoint {
         let mut t1 = self.square(); // 1.   t1 = u^2
         t1 *= Self::Z; // 2.   t1 = Z * t1              // Z * u^2
         let e1 = t1.ct_eq(&Self::MINUS_ONE); // 3.   e1 = t1 == -1            // exceptional case: Z * u^2 == -1
@@ -450,7 +464,26 @@ impl FieldElement {
         let mut y = y2.sqrt(); // 17.   y = sqrt(y2)
         let e3 = y.is_negative(); // 18.  e3 = sgn0(y) == 1
         y.conditional_negate(e2 ^ e3); //       y = CMOV(-y, y, e2 xor e3)
-        AffinePoint { x, y }
+        AffineMontgomeryPoint::new(x, y)
+    }
+
+    // See https://www.rfc-editor.org/rfc/rfc9380.html#name-curve448-q-3-mod-4-k-1.
+    // Without y-coordinate.
+    pub(crate) fn map_to_curve_elligator2_curve448_x(&self) -> FieldElement {
+        let mut t1 = self.square(); // 1.   t1 = u^2
+        t1 *= Self::Z; // 2.   t1 = Z * t1              // Z * u^2
+        let e1 = t1.ct_eq(&Self::MINUS_ONE); // 3.   e1 = t1 == -1            // exceptional case: Z * u^2 == -1
+        t1.conditional_assign(&Self::ZERO, e1); // 4.   t1 = CMOV(t1, 0, e1)     // if t1 == -1, set t1 = 0
+        let mut x1 = t1 + Self::ONE; // 5.   x1 = t1 + 1
+        x1 = x1.invert(); // 6.   x1 = inv0(x1)
+        x1 *= -Self::J; // 7.   x1 = -A * x1             // x1 = -A / (1 + Z * u^2)
+        let mut gx1 = x1 + Self::J; // 8.  gx1 = x1 + A
+        gx1 *= x1; // 9.  gx1 = gx1 * x1
+        gx1 += Self::ONE; // 10. gx1 = gx1 + B
+        gx1 *= x1; // 11. gx1 = gx1 * x1            // gx1 = x1^3 + A * x1^2 + B * x1
+        let x2 = -x1 - Self::J; // 12.  x2 = -x1 - A
+        let e2 = gx1.is_square(); // 14.  e2 = is_square(gx1)
+        Self::conditional_select(&x2, &x1, e2) // 15.   x = CMOV(x2, x1, e2)    // If is_square(gx1), x = x1, else x = x2
     }
 
     // See https://www.shiftleft.org/papers/decaf/decaf.pdf#section.A.3.
@@ -527,15 +560,13 @@ mod tests {
             .unwrap();
             let mut data = Array::<u8, U84>::default();
             expander.fill_bytes(&mut data).unwrap();
-            // TODO: This should be `Curve448FieldElement`.
             let u0 = FieldElement::reduce(&data);
             let mut e_u0 = *expected_u0;
             e_u0.reverse();
             let mut e_u1 = *expected_u1;
             e_u1.reverse();
             assert_eq!(u0.to_bytes(), e_u0);
             expander.fill_bytes(&mut data).unwrap();
-            // TODO: This should be `Curve448FieldElement`.
             let u1 = FieldElement::reduce(&data);
             assert_eq!(u1.to_bytes(), e_u1);
         }