我们已经看到了椭圆曲线上整数的操作，令人惊讶的是，我们可以将这些整数转换为有限域中的成员而不出现问题。请记住，字段成员的运算“+”和“.”是正常的算术运算，但结果会进行模运算。结果表明，即使基于模数操作，椭圆曲线点加法仍然成立。
例如，我们可以验证有限域的阶为103，从该字段中取出两个成员17和64，将它们组合为一个点坐标(17, 64)，该点在模103的曲线 y^2 = x^3 + 7 上：
y^2 = 64^2 % 103 = 79，x^3 + 7 = (17^3 + 7) % 103 = 79
现在让我们将Point结构重写，将其组件从big.Int改为FieldElement，修改后的代码如下所示：
package elliptic_curve
import (
"fmt"
"math/big"
)
type OP_TYPE int
const (
ADD OP_TYPE = iota
SUB
MUL
DIV
EXP
)
type Point struct {
a *FieldElement
b *FieldElement
x *FieldElement
y *FieldElement
}
func OpOnBig(x *FieldElement, y *FieldElement, scalar *big.Int, opType OP_TYPE) *FieldElement {
switch opType {
case ADD:
return x.Add(y)
case SUB:
return x.Subtract(y)
case MUL:
/*
Multiply in two cases, one is two field member multiply,
the other is field member multiply with scalar
*/
if y != nil {
return x.Multiply(y)
}
if scalar != nil {
return x.ScalarMul(scalar)
}
panic("error in multiply")
case DIV:
return x.Divide(y)
case EXP:
if scalar == nil {
panic("scalar should not be nil for EXP op")
}
return x.Power(scalar)
}
panic("should not come to here")
}
func NewEllipticCurvePoint(x *FieldElement, y *FieldElement, a *FieldElement, b *FieldElement) *Point {
if x == nil && y == nil {
return &Point{
x: x,
y: y,
a: a,
b: b,
}
}
left := OpOnBig(y, nil, big.NewInt(int64(2)), EXP)
x3 := OpOnBig(x, nil, big.NewInt(int64(3)), EXP)
ax := OpOnBig(a, x, nil, MUL)
right := OpOnBig(OpOnBig(x3, ax, nil, ADD), b, nil, ADD)
if left.EqualTo(right) != true {
err := fmt.Sprintf("Point(%v, %v) is not on the curve with a:%v, b:%v\n", x, y, a, b)
panic(err)
}
return &Point{
x: x,
y: y,
a: a,
b: b,
}
}
func (p *Point) Add(other *Point) *Point {
//check two points are on the same curve
if p.a.EqualTo(other.a) != true || p.b.EqualTo(other.b) != true {
panic("given two points are not on the same curve")
}
if p.x == nil {
return other
}
if other.x == nil {
return p
}
//points are on the vertical A(x,y) B(x,-y),
zero := NewFieldElement(p.x.order, big.NewInt(int64(0)))
if p.x.EqualTo(other.x) == true &&
OpOnBig(p.y, other.y, nil, ADD).EqualTo(zero) == true {
return &Point{
x: nil,
y: nil,
a: p.a,
b: p.b,
}
}
//find slope of line AB
//x1 -> p.x, y1->p.y, x2 ->other.x, y2->other.y
var numerator *FieldElement
var denominator *FieldElement
if p.x.EqualTo(other.x) == true && p.y.EqualTo(other.y) == true {
//slope = (3*x^2+a)/2y
xSqrt := OpOnBig(p.x, nil, big.NewInt(int64(2)), EXP)
threeXSqrt := OpOnBig(xSqrt, nil, big.NewInt(int64(3)), MUL)
numerator = OpOnBig(threeXSqrt, p.a, nil, ADD)
//denominator: 2y
denominator = OpOnBig(p.y, nil, big.NewInt(int64(2)), MUL)
} else {
numerator = OpOnBig(other.y, p.y, nil, SUB) //(y2-y1)
denominator = OpOnBig(other.x, p.x, nil, SUB) //(x2-x1)
}
//s=(y2-y1) / (x2-x1)
slope := OpOnBig(numerator, denominator, nil, DIV)
//s^2
slopeSqrt := OpOnBig(slope, nil, big.NewInt(int64(2)), EXP)
//x3 = s^2 - x1 - x2
x3 := OpOnBig(OpOnBig(slopeSqrt, p.x, nil, SUB), other.x, nil, SUB)
// x3-x1
x3Minusx1 := OpOnBig(x3, p.x, nil, SUB)
//y3 = s(x3-x1)+y1
y3 := OpOnBig(OpOnBig(slope, x3Minusx1, nil, MUL), p.y, nil, ADD)
//-y3
minusY3 := OpOnBig(y3, nil, big.NewInt(int64(-1)), MUL)
return &Point{
x: x3,
y: minusY3,
a: p.a,
b: p.b,
}
}
func (p *Point) String() string {
xString := "nil"
yString := "nil"
if p.x != nil {
xString = p.x.String()
}
if p.y != nil {
yString = p.y.String()
}
return fmt.Sprintf("(x:%s\n, y:%s\n, a:%s\n, b:%s\n)", xString, yString,
p.a.String(), p.b.String())
}
func (p *Point) Equal(other *Point) bool {
if p.a.EqualTo(other.a) == true && p.b.EqualTo(other.b) == true &&
p.x.EqualTo(other.x) == true &&
p.y.EqualTo(other.y) == true {
return true
}
return false
}
func (p *Point) NotEqual(other *Point) bool {
if p.a.EqualTo(other.a) != true || p.b.EqualTo(other.b) != true ||
p.x.EqualTo(other.x) != true ||
p.y.EqualTo(other.y) != true {
return true
}
return false
}
我们相应地修改了代码，没有新的内容，现在我们可以测试更改后的代码如下：
elliptic curve point over finite field is (x:FieldElement{order: 223, num: 192}
, y:FieldElement{order: 223, num: 105}
, a:FieldElement{order: 223, num: 0}
, b:FieldElement{order: 223, num: 7}
)
addition of points on vertical line over finit field is (x:nil
, y:nil
, a:FieldElement{order: 223, num: 0}
, b:FieldElement{order: 223, num: 7}
)
p1 + p2 over field order 223 is (x:FieldElement{order: 223, num: 170}
, y:FieldElement{order: 223, num: 142}
, a:FieldElement{order: 223, num: 0}
, b:FieldElement{order: 223, num: 7}
)
p1 + p1 over field order 223 is (x:FieldElement{order: 223, num: 49}
, y:FieldElement{order: 223, num: 71}
, a:FieldElement{order: 223, num: 0}
, b:FieldElement{order: 223, num: 7}
)
运行代码，我们得到结果：
elliptic curve point over finite field is (x:FieldElement{order: 223, num: 192}
, y:FieldElement{order: 223, num: 105}
, a:FieldElement{order: 223, num: 0}
, b:FieldElement{order: 223, num: 7}
)
addition of points on vertical line over finit field is (x:nil
, y:nil
, a:FieldElement{order: 223, num: 0}
, b:FieldElement{order: 223, num: 7}
)
p1 + p2 over field order 223 is (x:FieldElement{order: 223, num: 170}
, y:FieldElement{order: 223, num: 142}
, a:FieldElement{order: 223, num: 0}
, b:FieldElement{order: 223, num: 7}
)
p1 + p1 over field order 223 is (x:FieldElement{order: 223, num: 49}
, y:FieldElement{order: 223, num: 71}
, a:FieldElement{order: 223, num: 0}
, b:FieldElement{order: 223, num: 7}
)
请放心，我已经用纸和笔验证了这些结果，您可以相信我！
完整代码下载：
https://github.com/wycl16514/golang-bitcoin-elliptic-curve-cryptography
更多内容请参看：
http://m.study.163.com/provider/7600199/index.htm?share=2&shareId=7600199
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