github.com/yinchengtsinghua/golang-Eos-dpos-Ethereum@v0.0.0-20190121132951-92cc4225ed8e/crypto/bn256/cloudflare/curve.go

//此源码被清华学神尹成大魔王专业翻译分析并修改
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package bn256

import (
	"math/big"
)

//curvePoint实现椭圆曲线y？=x？+3。积分保存在雅各布文中。
//表格，有效时t=z²。g_是gf（p）上该曲线的点集。
type curvePoint struct {
	x, y, z, t gfP
}

var curveB = newGFp(3)

//曲线发电机是G_的发电机。
var curveGen = &curvePoint{
	x: *newGFp(1),
	y: *newGFp(2),
	z: *newGFp(1),
	t: *newGFp(1),
}

func (c *curvePoint) String() string {
	c.MakeAffine()
	x, y := &gfP{}, &gfP{}
	montDecode(x, &c.x)
	montDecode(y, &c.y)
	return "(" + x.String() + ", " + y.String() + ")"
}

func (c *curvePoint) Set(a *curvePoint) {
	c.x.Set(&a.x)
	c.y.Set(&a.y)
	c.z.Set(&a.z)
	c.t.Set(&a.t)
}

//is on curve返回真的iff c在曲线上。
func (c *curvePoint) IsOnCurve() bool {
	c.MakeAffine()
	if c.IsInfinity() {
		return true
	}

	y2, x3 := &gfP{}, &gfP{}
	gfpMul(y2, &c.y, &c.y)
	gfpMul(x3, &c.x, &c.x)
	gfpMul(x3, x3, &c.x)
	gfpAdd(x3, x3, curveB)

	return *y2 == *x3
}

func (c *curvePoint) SetInfinity() {
	c.x = gfP{0}
	c.y = *newGFp(1)
	c.z = gfP{0}
	c.t = gfP{0}
}

func (c *curvePoint) IsInfinity() bool {
	return c.z == gfP{0}
}

func (c *curvePoint) Add(a, b *curvePoint) {
	if a.IsInfinity() {
		c.Set(b)
		return
	}
	if b.IsInfinity() {
		c.Set(a)
		return
	}

//见http://hyper椭圆形.org/efd/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3

//通过替换a=[x1:y1:z1]和b=[x2:y2:z2]来规范化点。
//通过[U1:S1:Z1·Z2]和[U2:S2:Z1·Z2]
//式中：U1=x1·z2？、S1=y1·z2？、U1=x2·z1？、S2=y2·z1？？
	z12, z22 := &gfP{}, &gfP{}
	gfpMul(z12, &a.z, &a.z)
	gfpMul(z22, &b.z, &b.z)

	u1, u2 := &gfP{}, &gfP{}
	gfpMul(u1, &a.x, z22)
	gfpMul(u2, &b.x, z12)

	t, s1 := &gfP{}, &gfP{}
	gfpMul(t, &b.z, z22)
	gfpMul(s1, &a.y, t)

	s2 := &gfP{}
	gfpMul(t, &a.z, z12)
	gfpMul(s2, &b.y, t)

//计算x=（2h）2（s²-u1-u2）
//其中s=（s2-s1）/（u2-u1）是线路通过的坡度
//（U1、S1）和（U2、S2）。额外因子2h=2（u2-u1）来自下面的z值。
//这也是：
//4（s2-s1）2-4h 2（u1+u2）=4（s2-s1）2-4h 3-4h 2（2u1）
//R＝J-2V
//以及下面的注释。
	h := &gfP{}
	gfpSub(h, u2, u1)
	xEqual := *h == gfP{0}

	gfpAdd(t, h, h)
//I= 4H
	i := &gfP{}
	gfpMul(i, t, t)
//J= 4H
	j := &gfP{}
	gfpMul(j, h, i)

	gfpSub(t, s2, s1)
	yEqual := *t == gfP{0}
	if xEqual && yEqual {
		c.Double(a)
		return
	}
	r := &gfP{}
	gfpAdd(r, t, t)

	v := &gfP{}
	gfpMul(v, u1, i)

//T4＝4（S2-S1）
	t4, t6 := &gfP{}, &gfP{}
	gfpMul(t4, r, r)
	gfpAdd(t, v, v)
	gfpSub(t6, t4, j)

	gfpSub(&c.x, t6, t)

//设置y=—（2h）³（s1+s*（x/4h²-u1））
//这也是
//y=-2·s1·j-（s2-s1）（2x-2i·u1）=r（v-x）-2·s1·j
gfpSub(t, v, &c.x) //T7
gfpMul(t4, s1, j)  //T8
gfpAdd(t6, t4, t4) //T9
gfpMul(t4, r, t)   //T10
	gfpSub(&c.y, t4, t6)

//设置z=2（u2-u1）·z1·z2=2h·z1·z2
gfpAdd(t, &a.z, &b.z) //T11
gfpMul(t4, t, t)      //T12
gfpSub(t, t4, z12)    //T13
gfpSub(t4, t, z22)    //T14
	gfpMul(&c.z, t4, h)
}

func (c *curvePoint) Double(a *curvePoint) {
//请参阅http://hyper椭圆形.org/efd/g1p/auto-code/shortw/jacobian-0/double/dbl-2009-l.op3
	A, B, C := &gfP{}, &gfP{}, &gfP{}
	gfpMul(A, &a.x, &a.x)
	gfpMul(B, &a.y, &a.y)
	gfpMul(C, B, B)

	t, t2 := &gfP{}, &gfP{}
	gfpAdd(t, &a.x, B)
	gfpMul(t2, t, t)
	gfpSub(t, t2, A)
	gfpSub(t2, t, C)

	d, e, f := &gfP{}, &gfP{}, &gfP{}
	gfpAdd(d, t2, t2)
	gfpAdd(t, A, A)
	gfpAdd(e, t, A)
	gfpMul(f, e, e)

	gfpAdd(t, d, d)
	gfpSub(&c.x, f, t)

	gfpAdd(t, C, C)
	gfpAdd(t2, t, t)
	gfpAdd(t, t2, t2)
	gfpSub(&c.y, d, &c.x)
	gfpMul(t2, e, &c.y)
	gfpSub(&c.y, t2, t)

	gfpMul(t, &a.y, &a.z)
	gfpAdd(&c.z, t, t)
}

func (c *curvePoint) Mul(a *curvePoint, scalar *big.Int) {
	precomp := [1 << 2]*curvePoint{nil, {}, {}, {}}
	precomp[1].Set(a)
	precomp[2].Set(a)
	gfpMul(&precomp[2].x, &precomp[2].x, xiTo2PSquaredMinus2Over3)
	precomp[3].Add(precomp[1], precomp[2])

	multiScalar := curveLattice.Multi(scalar)

	sum := &curvePoint{}
	sum.SetInfinity()
	t := &curvePoint{}

	for i := len(multiScalar) - 1; i >= 0; i-- {
		t.Double(sum)
		if multiScalar[i] == 0 {
			sum.Set(t)
		} else {
			sum.Add(t, precomp[multiScalar[i]])
		}
	}
	c.Set(sum)
}

func (c *curvePoint) MakeAffine() {
	if c.z == *newGFp(1) {
		return
	} else if c.z == *newGFp(0) {
		c.x = gfP{0}
		c.y = *newGFp(1)
		c.t = gfP{0}
		return
	}

	zInv := &gfP{}
	zInv.Invert(&c.z)

	t, zInv2 := &gfP{}, &gfP{}
	gfpMul(t, &c.y, zInv)
	gfpMul(zInv2, zInv, zInv)

	gfpMul(&c.x, &c.x, zInv2)
	gfpMul(&c.y, t, zInv2)

	c.z = *newGFp(1)
	c.t = *newGFp(1)
}

func (c *curvePoint) Neg(a *curvePoint) {
	c.x.Set(&a.x)
	gfpNeg(&c.y, &a.y)
	c.z.Set(&a.z)
	c.t = gfP{0}
}