// Copyright 2012 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package bn256
import (
"math/big"
)
// twistPoint implements the elliptic curve y²=x³+3/ξ over GF(p²). Points are
// kept in Jacobian form and t=z² when valid. The group G₂ is the set of
// n-torsion points of this curve over GF(p²) (where n = Order)
type twistPoint struct {
x, y, z, t *gfP2
}
var twistB = &gfP2{
bigFromBase10("266929791119991161246907387137283842545076965332900288569378510910307636690"),
bigFromBase10("19485874751759354771024239261021720505790618469301721065564631296452457478373"),
}
// twistGen is the generator of group G₂.
var twistGen = &twistPoint{
&gfP2{
bigFromBase10("11559732032986387107991004021392285783925812861821192530917403151452391805634"),
bigFromBase10("10857046999023057135944570762232829481370756359578518086990519993285655852781"),
},
&gfP2{
bigFromBase10("4082367875863433681332203403145435568316851327593401208105741076214120093531"),
bigFromBase10("8495653923123431417604973247489272438418190587263600148770280649306958101930"),
},
&gfP2{
bigFromBase10("0"),
bigFromBase10("1"),
},
&gfP2{
bigFromBase10("0"),
bigFromBase10("1"),
},
}
func newTwistPoint(pool *bnPool) *twistPoint {
return &twistPoint{
newGFp2(pool),
newGFp2(pool),
newGFp2(pool),
newGFp2(pool),
}
}
func (c *twistPoint) String() string {
return "(" + c.x.String() + ", " + c.y.String() + ", " + c.z.String() + ")"
}
func (c *twistPoint) Put(pool *bnPool) {
c.x.Put(pool)
c.y.Put(pool)
c.z.Put(pool)
c.t.Put(pool)
}
func (c *twistPoint) Set(a *twistPoint) {
c.x.Set(a.x)
c.y.Set(a.y)
c.z.Set(a.z)
c.t.Set(a.t)
}
// IsOnCurve returns true iff c is on the curve where c must be in affine form.
func (c *twistPoint) IsOnCurve() bool {
pool := new(bnPool)
yy := newGFp2(pool).Square(c.y, pool)
xxx := newGFp2(pool).Square(c.x, pool)
xxx.Mul(xxx, c.x, pool)
yy.Sub(yy, xxx)
yy.Sub(yy, twistB)
yy.Minimal()
return yy.x.Sign() == 0 && yy.y.Sign() == 0
}
func (c *twistPoint) SetInfinity() {
c.z.SetZero()
}
func (c *twistPoint) IsInfinity() bool {
return c.z.IsZero()
}
func (c *twistPoint) Add(a, b *twistPoint, pool *bnPool) {
// For additional comments, see the same function in curve.go.
if a.IsInfinity() {
c.Set(b)
return
}
if b.IsInfinity() {
c.Set(a)
return
}
// See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3
z1z1 := newGFp2(pool).Square(a.z, pool)
z2z2 := newGFp2(pool).Square(b.z, pool)
u1 := newGFp2(pool).Mul(a.x, z2z2, pool)
u2 := newGFp2(pool).Mul(b.x, z1z1, pool)
t := newGFp2(pool).Mul(b.z, z2z2, pool)
s1 := newGFp2(pool).Mul(a.y, t, pool)
t.Mul(a.z, z1z1, pool)
s2 := newGFp2(pool).Mul(b.y, t, pool)
h := newGFp2(pool).Sub(u2, u1)
xEqual := h.IsZero()
t.Add(h, h)
i := newGFp2(pool).Square(t, pool)
j := newGFp2(pool).Mul(h, i, pool)
t.Sub(s2, s1)
yEqual := t.IsZero()
if xEqual && yEqual {
c.Double(a, pool)
return
}
r := newGFp2(pool).Add(t, t)
v := newGFp2(pool).Mul(u1, i, pool)
t4 := newGFp2(pool).Square(r, pool)
t.Add(v, v)
t6 := newGFp2(pool).Sub(t4, j)
c.x.Sub(t6, t)
t.Sub(v, c.x) // t7
t4.Mul(s1, j, pool) // t8
t6.Add(t4, t4) // t9
t4.Mul(r, t, pool) // t10
c.y.Sub(t4, t6)
t.Add(a.z, b.z) // t11
t4.Square(t, pool) // t12
t.Sub(t4, z1z1) // t13
t4.Sub(t, z2z2) // t14
c.z.Mul(t4, h, pool)
z1z1.Put(pool)
z2z2.Put(pool)
u1.Put(pool)
u2.Put(pool)
t.Put(pool)
s1.Put(pool)
s2.Put(pool)
h.Put(pool)
i.Put(pool)
j.Put(pool)
r.Put(pool)
v.Put(pool)
t4.Put(pool)
t6.Put(pool)
}
func (c *twistPoint) Double(a *twistPoint, pool *bnPool) {
// See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
A := newGFp2(pool).Square(a.x, pool)
B := newGFp2(pool).Square(a.y, pool)
C := newGFp2(pool).Square(B, pool)
t := newGFp2(pool).Add(a.x, B)
t2 := newGFp2(pool).Square(t, pool)
t.Sub(t2, A)
t2.Sub(t, C)
d := newGFp2(pool).Add(t2, t2)
t.Add(A, A)
e := newGFp2(pool).Add(t, A)
f := newGFp2(pool).Square(e, pool)
t.Add(d, d)
c.x.Sub(f, t)
t.Add(C, C)
t2.Add(t, t)
t.Add(t2, t2)
c.y.Sub(d, c.x)
t2.Mul(e, c.y, pool)
c.y.Sub(t2, t)
t.Mul(a.y, a.z, pool)
c.z.Add(t, t)
A.Put(pool)
B.Put(pool)
C.Put(pool)
t.Put(pool)
t2.Put(pool)
d.Put(pool)
e.Put(pool)
f.Put(pool)
}
func (c *twistPoint) Mul(a *twistPoint, scalar *big.Int, pool *bnPool) *twistPoint {
sum := newTwistPoint(pool)
sum.SetInfinity()
t := newTwistPoint(pool)
for i := scalar.BitLen(); i >= 0; i-- {
t.Double(sum, pool)
if scalar.Bit(i) != 0 {
sum.Add(t, a, pool)
} else {
sum.Set(t)
}
}
c.Set(sum)
sum.Put(pool)
t.Put(pool)
return c
}
func (c *twistPoint) MakeAffine(pool *bnPool) *twistPoint {
if c.z.IsOne() {
return c
}
zInv := newGFp2(pool).Invert(c.z, pool)
t := newGFp2(pool).Mul(c.y, zInv, pool)
zInv2 := newGFp2(pool).Square(zInv, pool)
c.y.Mul(t, zInv2, pool)
t.Mul(c.x, zInv2, pool)
c.x.Set(t)
c.z.SetOne()
c.t.SetOne()
zInv.Put(pool)
t.Put(pool)
zInv2.Put(pool)
return c
}
func (c *twistPoint) Negative(a *twistPoint, pool *bnPool) {
c.x.Set(a.x)
c.y.SetZero()
c.y.Sub(c.y, a.y)
c.z.Set(a.z)
c.t.SetZero()
}
