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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{
gfP{0x75046774386b8d71, 0x5bd0854a46d36cf8, 0x664327a1d41c8414, 0x96c9abb932eeb2f},
gfP{0xb94f760fb4c5ee14, 0xdae9f8f24c3b6eb4, 0x77a675d2e52f4fe4, 0x736f31b09116c66b},
}
// twistGen is the generator of group G₂.
var twistGen = &twistPoint{
gfP2{
gfP{0x402c4ab7139e1404, 0xce1c368a183d85a4, 0xd67cf9a6cb8d3983, 0x3cf246bbc2a9fbe8},
gfP{0x88f9f11da7cdc184, 0x18293f95d69509d3, 0xb5ce0c55a735d5a1, 0x15134189bfd45a0},
},
gfP2{
gfP{0xbfac7d731e9e87a2, 0xa50bb8007962e441, 0xafe910a4e8270556, 0x5075c5429d69159a},
gfP{0xc2e07c1463ea9e56, 0xee4442052072ebd2, 0x561a519486036937, 0x5bd9394cc0d2cce},
},
gfP2{*newGFp(0), *newGFp(1)},
gfP2{*newGFp(0), *newGFp(1)},
}
func (c *twistPoint) String() string {
c.MakeAffine()
x, y := gfP2Decode(&c.x), gfP2Decode(&c.y)
return "(" + x.String() + ", " + y.String() + ")"
}
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.
func (c *twistPoint) IsOnCurve() bool {
c.MakeAffine()
if c.IsInfinity() {
return true
}
y2, x3 := &gfP2{}, &gfP2{}
y2.Square(&c.y)
x3.Square(&c.x).Mul(x3, &c.x).Add(x3, twistB)
return *y2 == *x3
}
func (c *twistPoint) SetInfinity() {
c.x.SetZero()
c.y.SetOne()
c.z.SetZero()
c.t.SetZero()
}
func (c *twistPoint) IsInfinity() bool {
return c.z.IsZero()
}
func (c *twistPoint) Add(a, b *twistPoint) {
// 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
z12 := (&gfP2{}).Square(&a.z)
z22 := (&gfP2{}).Square(&b.z)
u1 := (&gfP2{}).Mul(&a.x, z22)
u2 := (&gfP2{}).Mul(&b.x, z12)
t := (&gfP2{}).Mul(&b.z, z22)
s1 := (&gfP2{}).Mul(&a.y, t)
t.Mul(&a.z, z12)
s2 := (&gfP2{}).Mul(&b.y, t)
h := (&gfP2{}).Sub(u2, u1)
xEqual := h.IsZero()
t.Add(h, h)
i := (&gfP2{}).Square(t)
j := (&gfP2{}).Mul(h, i)
t.Sub(s2, s1)
yEqual := t.IsZero()
if xEqual && yEqual {
c.Double(a)
return
}
r := (&gfP2{}).Add(t, t)
v := (&gfP2{}).Mul(u1, i)
t4 := (&gfP2{}).Square(r)
t.Add(v, v)
t6 := (&gfP2{}).Sub(t4, j)
c.x.Sub(t6, t)
t.Sub(v, &c.x) // t7
t4.Mul(s1, j) // t8
t6.Add(t4, t4) // t9
t4.Mul(r, t) // t10
c.y.Sub(t4, t6)
t.Add(&a.z, &b.z) // t11
t4.Square(t) // t12
t.Sub(t4, z12) // t13
t4.Sub(t, z22) // t14
c.z.Mul(t4, h)
}
func (c *twistPoint) Double(a *twistPoint) {
// See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
A := (&gfP2{}).Square(&a.x)
B := (&gfP2{}).Square(&a.y)
C := (&gfP2{}).Square(B)
t := (&gfP2{}).Add(&a.x, B)
t2 := (&gfP2{}).Square(t)
t.Sub(t2, A)
t2.Sub(t, C)
d := (&gfP2{}).Add(t2, t2)
t.Add(A, A)
e := (&gfP2{}).Add(t, A)
f := (&gfP2{}).Square(e)
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)
c.y.Sub(t2, t)
t.Mul(&a.y, &a.z)
c.z.Add(t, t)
}
func (c *twistPoint) Mul(a *twistPoint, scalar *big.Int) {
sum, t := &twistPoint{}, &twistPoint{}
for i := scalar.BitLen(); i >= 0; i-- {
t.Double(sum)
if scalar.Bit(i) != 0 {
sum.Add(t, a)
} else {
sum.Set(t)
}
}
c.Set(sum)
}
func (c *twistPoint) MakeAffine() {
if c.z.IsOne() {
return
} else if c.z.IsZero() {
c.x.SetZero()
c.y.SetOne()
c.t.SetZero()
return
}
zInv := (&gfP2{}).Invert(&c.z)
t := (&gfP2{}).Mul(&c.y, zInv)
zInv2 := (&gfP2{}).Square(zInv)
c.y.Mul(t, zInv2)
t.Mul(&c.x, zInv2)
c.x.Set(t)
c.z.SetOne()
c.t.SetOne()
}
func (c *twistPoint) Neg(a *twistPoint) {
c.x.Set(&a.x)
c.y.Neg(&a.y)
c.z.Set(&a.z)
c.t.SetZero()
}