Optimize Edwards to Montgomery conversions

ed448-goldilocks/src/edwards/affine.rs

@@ -110,6 +110,18 @@ impl AffinePoint {
         }
     }
 
+    /// Convert this point to [`MontgomeryPoint`]
+    // See https://www.rfc-editor.org/rfc/rfc7748#section-4.2 4-isogeny maps
+    pub fn to_montgomery(&self) -> MontgomeryPoint {
+        // u = y^2/x^2
+
+        // Simplified to:
+        // u = (y/x)^2
+        let u = (self.y * self.x.invert()).square();
+
+        MontgomeryPoint(u.to_bytes())
+    }
+
     /// The X coordinate
     pub fn x(&self) -> [u8; 56] {
         self.x.to_bytes()

ed448-goldilocks/src/edwards/extended.rs

@@ -529,16 +529,19 @@ impl EdwardsPoint {
     };
 
     /// Convert this point to [`MontgomeryPoint`]
+    // See https://www.rfc-editor.org/rfc/rfc7748#section-4.2 4-isogeny maps
     pub fn to_montgomery(&self) -> MontgomeryPoint {
-        // u = y^2 * [(1-dy^2)/(1-y^2)]
+        // Affine map:
+        // (x, y) = (x/z, y/z)
+        // Edwards to montgomery:
+        // u = y^2/x^2
 
-        let affine = self.to_affine();
-
-        let yy = affine.y.square();
-        let dyy = FieldElement::EDWARDS_D * yy;
-
-        let u = yy * (FieldElement::ONE - dyy) * (FieldElement::ONE - yy).invert();
+        // Simplified to:
+        // u = (y/z)^2/(x/z)^2
+        // u = ((y/z)/(x/z))^2
+        // u = (y/x)^2
 
+        let u = (self.Y * self.X.invert()).square();
         MontgomeryPoint(u.to_bytes())
     }
 

ed448-goldilocks/src/montgomery.rs

@@ -233,4 +233,18 @@ mod tests {
         let goldilocks_point = bp.scalar_mul(&scalar);
         assert_eq!(goldilocks_point.to_montgomery(), montgomery_res);
     }
+
+    #[test]
+    fn test_montgomery_edwards_affine() {
+        let scalar = EdwardsScalar::from(200u32);
+        use crate::GOLDILOCKS_BASE_POINT as bp;
+
+        // Montgomery scalar mul
+        let montgomery_bp = bp.to_affine().to_montgomery();
+        let montgomery_res = &montgomery_bp * &scalar;
+
+        // Goldilocks scalar mul
+        let goldilocks_point = bp.scalar_mul(&scalar).to_affine();
+        assert_eq!(goldilocks_point.to_montgomery(), montgomery_res);
+    }
 }