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addition on twisted Edwards #267

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Feb 18, 2022
Merged

addition on twisted Edwards #267

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42 changes: 24 additions & 18 deletions std/algebra/twistededwards/bandersnatch/point.go
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Original file line number Diff line number Diff line change
Expand Up @@ -57,24 +57,30 @@ func (p *Point) MustBeOnCurve(api frontend.API, curve EdCurve) {
// p1, p2, c are respectively: the point to add, a known base point, and the parameters of the twisted edwards curve
func (p *Point) Add(api frontend.API, p1, p2 *Point, curve EdCurve) *Point {

// https://eprint.iacr.org/2008/013.pdf

n11 := api.Mul(p1.X, p2.Y)
n12 := api.Mul(p1.Y, p2.X)
n1 := api.Add(n11, n12)

n21 := api.Mul(p1.Y, p2.Y)
n22 := api.Mul(p1.X, p2.X)
an22 := api.Mul(n22, &curve.A)
n2 := api.Sub(n21, an22)

d11 := api.Mul(curve.D, n11, n12)
d1 := api.Add(1, d11)

d2 := api.Sub(1, d11)

p.X = api.DivUnchecked(n1, d1)
p.Y = api.DivUnchecked(n2, d2)
// u = (x1 + y1) * (x2 + y2)
u1 := api.Mul(p1.X, &curve.A)
u1 = api.Sub(p1.Y, u1)
u2 := api.Add(p2.X, p2.Y)
u := api.Mul(u1, u2)

// v0 = x1 * y2
v0 := api.Mul(p2.Y, p1.X)

// v1 = x2 * y1
v1 := api.Mul(p2.X, p1.Y)

// v2 = d * v0 * v1
v2 := api.Mul(&curve.D, v0, v1)

// x = (v0 + v1) / (1 + v2)
p.X = api.Add(v0, v1)
p.X = api.DivUnchecked(p.X, api.Add(1, v2))

// y = (u + a * v0 - v1) / (1 - v2)
p.Y = api.Mul(&curve.A, v0)
p.Y = api.Sub(p.Y, v1)
p.Y = api.Add(p.Y, u)
p.Y = api.DivUnchecked(p.Y, api.Sub(1, v2))

return p
}
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42 changes: 24 additions & 18 deletions std/algebra/twistededwards/point.go
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Original file line number Diff line number Diff line change
Expand Up @@ -57,24 +57,30 @@ func (p *Point) MustBeOnCurve(api frontend.API, curve EdCurve) {
// p1, p2, c are respectively: the point to add, a known base point, and the parameters of the twisted edwards curve
func (p *Point) Add(api frontend.API, p1, p2 *Point, curve EdCurve) *Point {

// https://eprint.iacr.org/2008/013.pdf

n11 := api.Mul(p1.X, p2.Y)
n12 := api.Mul(p1.Y, p2.X)
n1 := api.Add(n11, n12)

n21 := api.Mul(p1.Y, p2.Y)
n22 := api.Mul(p1.X, p2.X)
an22 := api.Mul(n22, &curve.A)
n2 := api.Sub(n21, an22)

d11 := api.Mul(curve.D, n11, n12)
d1 := api.Add(1, d11)

d2 := api.Sub(1, d11)

p.X = api.DivUnchecked(n1, d1)
p.Y = api.DivUnchecked(n2, d2)
// u = (x1 + y1) * (x2 + y2)
u1 := api.Mul(p1.X, &curve.A)
u1 = api.Sub(p1.Y, u1)
u2 := api.Add(p2.X, p2.Y)
u := api.Mul(u1, u2)

// v0 = x1 * y2
v0 := api.Mul(p2.Y, p1.X)

// v1 = x2 * y1
v1 := api.Mul(p2.X, p1.Y)

// v2 = d * v0 * v1
v2 := api.Mul(&curve.D, v0, v1)

// x = (v0 + v1) / (1 + v2)
p.X = api.Add(v0, v1)
p.X = api.DivUnchecked(p.X, api.Add(1, v2))

// y = (u + a * v0 - v1) / (1 - v2)
p.Y = api.Mul(&curve.A, v0)
p.Y = api.Sub(p.Y, v1)
p.Y = api.Add(p.Y, u)
p.Y = api.DivUnchecked(p.Y, api.Sub(1, v2))

return p
}
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