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Optimize Montgomery to Edwards conversions #1308

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Optimize Montgomery to Edwards conversions #1308

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69 changes: 41 additions & 28 deletions ed448-goldilocks/src/edwards/affine.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -86,35 +86,48 @@ impl AffinePoint {
}
}

// https://www.rfc-editor.org/rfc/rfc7748#section-4.2
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(),
}
let u = self.x;
let v = self.y;

// x = (4*v*(u^2 - 1)/(u^4 - 2*u^2 + 4*v^2 + 1)
// y = -(u^5 - 2*u^3 - 4*u*v^2 + u)/(u^5 - 2*u^2*v^2 - 2*u^3 - 2*v^2 + u))

let uu = u.square();
let uu_minus_1 = uu - FieldElement::ONE;
let uu_minus_1_sq = uu_minus_1.square();
let vv2 = v.square().double();
let vv4 = vv2.double();

// 4*v*(u^2 - 1)
let xn = v.double().double() * uu_minus_1;
// u^4 - 2*u^2 + 4*v^2 + 1
// Simplified to:
// = u^4 - 2*u^2 + 1 + 4*v^2
// = (u^2 - 1)^2 - 4*v^2 | (perfect square trinomial)
let xd = uu_minus_1_sq + vv4;

// -(u^5 - 2*u^3 - 4*u*v^2 + u)
// Simplified to:
// = -u * (u^4 - 2*u^2 - 4*v^2 + 1)
// = -u * (u^4 - 2*u^2 + 1 - 4*v^2)
// = -u * ((u^2 - 1)^2 - 4*v^2) | (perfect square trinomial)
let yn = -u * (uu_minus_1_sq - vv4);
// u^5 - 2*u^2*v^2 - 2*u^3 - 2*v^2 + u
// Simplified to:
// = u^5 - 2*u^3 + u - 2*u^2*v^2 - 2*v^2
// = u * (u^4 - 2*u^2 + 1) - 2*v^2 * (u^2 + 1)
// = u * (u^2 - 1)^2 - 2*v^2 * (u^2 + 1) | (perfect square trinomial)
let yd = u * uu_minus_1_sq - vv2 * (uu + FieldElement::ONE);

// Simplified two denominators to a single inversion.
let d = (xd * yd).invert();

let x = xn * yd * d;
let y = yn * xd * d;

Self { x, y }
}

/// Standard compression; store Y and sign of X
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