golang 实现比特币内核:处理椭圆曲线中的天文数字

tyler_download2024-11-08 12:29

在比特币密码学中，我们需要处理天文数字，这个数字是如此巨大，以至于它很容易超出我们宇宙中原子的总数，也许 64 位的值不足以表示这个数字，而像加、乘、幂这样的操作如果使用 64 位整数会导致溢出，因此我们可能需要借助 golang 的 big 包，我们将通过使用 big.Int 来表示其值字段来更改 FieldNumber 的代码，代码将如下所示：

go 复制代码 
package elliptic_curve

import (
	"fmt"
	"math/big"
)

//using big package to deal with Astronomical figures

type FieldElement struct {
	order *big.Int //field order
	num   *big.Int //value of the given element in the field
}

func NewFieldElement(order *big.Int, num *big.Int) *FieldElement {
	/*
		constructor for FieldElement, its the __init__ if you are from python
	*/
	if order.Cmp(num) == -1 {
		err := fmt.Sprintf("Num not in the range from 0 to %v", order)
		panic(err)
	}

	return &FieldElement{
		order: order,
		num:   num,
	}
}

func (f *FieldElement) String() string {
	//format the object to printable string
	//its __repr__ if you are from python
	return fmt.Sprintf("FieldElement{order: %v, num: %v}", *f.order, *f.num)
}

func (f *FieldElement) EqualTo(other *FieldElement) bool {
	/*
		two field element is equal if their order and value are equal
	*/
	return f.order.Cmp(other.order) == 0 && f.num.Cmp(other.num) == 0
}

func (f *FieldElement) checkOrder(other *FieldElement) {
	if f.order.Cmp(other.order) != 0 {
		panic("add need to do on field element with the same order")
	}
}

func (f *FieldElement) Add(other *FieldElement) *FieldElement {

	f.checkOrder(other)
	//remember to do the modulur
	var op big.Int
	return NewFieldElement(f.order, op.Mod(op.Add(f.num, other.num), f.order))
}

func (f *FieldElement) Negate() *FieldElement {
	/*
		for a field element a, its negate is another element b in field such that
		(a + b) % order= 0(remember the modulur over order), because the value of element
		in the field are smaller than its order, we can easily get the negate of a by
		order - a,
	*/
	var op big.Int
	return NewFieldElement(f.order, op.Sub(f.order, f.num))
}

func (f *FieldElement) Subtract(other *FieldElement) *FieldElement {
	//first find the negate of the other
	//add this and the negate of the other
	return f.Add(other.Negate())
}

func (f *FieldElement) Multiply(other *FieldElement) *FieldElement {
	f.checkOrder(other)
	//multiplie over modulur of order
	var op big.Int
	mul := op.Mul(f.num, other.num)
	return NewFieldElement(f.order, op.Mod(mul, f.order))
}

func (f *FieldElement) Power(power *big.Int) *FieldElement {
	var op big.Int
	powerRes := op.Exp(f.num, power, nil)
	modRes := op.Mod(powerRes, f.order)
	return NewFieldElement(f.order, modRes)
}

func (f *FieldElement) ScalarMul(val *big.Int) *FieldElement {
	var op big.Int
	res := op.Mul(f.num, val)
	res = op.Mod(res, f.order)
	return NewFieldElement(f.order, res)
}

现在我们需要确保这些更改不会破坏我们的逻辑，让我们再次运行测试，在 main.go 中，我们有以下代码：

go 复制代码 
package main

import (
	ecc "elliptic_curve"
	"fmt"
	"math/big"
	"math/rand"
)

func SolveField19MultiplieSet() {
	//randomly select a num from (1, 18)
	min := 1
	max := 18
	k := rand.Intn(max-min) + min
	fmt.Printf("randomly select k is : %d\n", k)
	element := ecc.NewFieldElement(big.NewInt(19), big.NewInt(int64(k)))
	for i := 0; i < 19; i++ {
		fmt.Printf("element %d multiplie with %d is %v\n", k, i,
			element.ScalarMul(big.NewInt(int64(i))))
	}

}

func main() {
	f44 := ecc.NewFieldElement(big.NewInt(57), big.NewInt(44))
	f33 := ecc.NewFieldElement(big.NewInt(57), big.NewInt(33))
	// 44 + 33 equal to (44+33) % 57 is 20
	res := f44.Add(f33)
	fmt.Printf("field element 44 add to field element 33 is : %v\n", res)
	//-44 is the negate of field element 44, which is 57 - 44 = 13
	fmt.Printf("negate of field element 44 is : %v\n", f44.Negate())

	fmt.Printf("field element 44 - 33 is : %v\n", f44.Subtract(f33))
	fmt.Printf("field element 33 - 44 is : %v\n", f33.Subtract(f44))

	//it is easy to check (11+33)%57 == 44
	//check (46 + 44) % 57 == 33
	fmt.Printf("check 46 + 44 over modulur 57 is %d\n", (46+44)%57)
	//check by field element
	f46 := ecc.NewFieldElement(big.NewInt(57), big.NewInt(46))
	fmt.Printf("field element 46 + 44 is %v\n", f46.Add(f44))

	SolveField19MultiplieSet()
}

运行上述代码将获得以下结果：

go 复制代码 
field element 44 add to field element 33 is : FieldElement{order: 57, num: 20}
negate of field element 44 is : FieldElement{order: 57, num: 13}
field element 44 - 33 is : FieldElement{order: 57, num: 11}
field element 33 - 44 is : FieldElement{order: 57, num: 46}
check 46 + 44 over modulur 57 is 33
field element 46 + 44 is FieldElement{order: 57, num: 33}
randomly select k is : 2
element 2 multiplie with 0 is FieldElement{order: 19, num: 0}
element 2 multiplie with 1 is FieldElement{order: 19, num: 2}
element 2 multiplie with 2 is FieldElement{order: 19, num: 4}
element 2 multiplie with 3 is FieldElement{order: 19, num: 6}
element 2 multiplie with 4 is FieldElement{order: 19, num: 8}
element 2 multiplie with 5 is FieldElement{order: 19, num: 10}
element 2 multiplie with 6 is FieldElement{order: 19, num: 12}
element 2 multiplie with 7 is FieldElement{order: 19, num: 14}
element 2 multiplie with 8 is FieldElement{order: 19, num: 16}
element 2 multiplie with 9 is FieldElement{order: 19, num: 18}
element 2 multiplie with 10 is FieldElement{order: 19, num: 1}
element 2 multiplie with 11 is FieldElement{order: 19, num: 3}
element 2 multiplie with 12 is FieldElement{order: 19, num: 5}
element 2 multiplie with 13 is FieldElement{order: 19, num: 7}
element 2 multiplie with 14 is FieldElement{order: 19, num: 9}
element 2 multiplie with 15 is FieldElement{order: 19, num: 11}
element 2 multiplie with 16 is FieldElement{order: 19, num: 13}
element 2 multiplie with 17 is FieldElement{order: 19, num: 15}
element 2 multiplie with 18 is FieldElement{order: 19, num: 17}

通过检查结果，我们可以确保 FieldElement 中的更改不会破坏我们之前的逻辑。现在让我们考虑以下问题：

p = 7, 11, 17, 19, 31，以下集合会是什么：

{1 ^(p-1), 2 ^ (p-1), ... (p-1)^(p-1)}

让我们在 main.go 中编写代码来解决它：

go 复制代码 
func ComputeFieldOrderPower() {
	orders := []int{7, 11, 17, 31}
	for _, p := range orders {
		fmt.Printf("value of p is: %d\n", p)
		for i := 1; i < p; i++ {
			elm := ecc.NewFieldElement(big.NewInt(int64(p)), big.NewInt(int64(i)))
			fmt.Printf("for element: %v, its power of p - 1 is: %v\n", elm,
				elm.Power(big.NewInt(int64(p-1))))
		}
		fmt.Println("-------------------------------")
	}
}

func main() {
    ComputeFieldOrderPower()
}

结果如下：

go 复制代码 
value of p is: 7
for element: FieldElement{order: 7, num: 1}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 2}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 3}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 4}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 5}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 6}, its power of p - 1 is: FieldElement{order: 7, num: 1}
-------------------------------
value of p is: 11
for element: FieldElement{order: 11, num: 1}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 2}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 3}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 4}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 5}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 6}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 7}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 8}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 9}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 10}, its power of p - 1 is: FieldElement{order: 11, num: 1}
-------------------------------
value of p is: 17
for element: FieldElement{order: 17, num: 1}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 2}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 3}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 4}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 5}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 6}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 7}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 8}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 9}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 10}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 11}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 12}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 13}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 14}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 15}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 16}, its power of p - 1 is: FieldElement{order: 17, num: 1}
-------------------------------
value of p is: 31
for element: FieldElement{order: 31, num: 1}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 2}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 3}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 4}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 5}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 6}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 7}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 8}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 9}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 10}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 11}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 12}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 13}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 14}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 15}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 16}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 17}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 18}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 19}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 20}, its power of p - 1 is: FieldElement{order: 31, num: 1}
my@MACdeMacBook-Air bitcoin % go run main.go
value of p is: 7
for element: FieldElement{order: 7, num: 1}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 2}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 3}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 4}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 5}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 6}, its power of p - 1 is: FieldElement{order: 7, num: 1}
-------------------------------
value of p is: 11
for element: FieldElement{order: 11, num: 1}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 2}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 3}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 4}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 5}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 6}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 7}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 8}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 9}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 10}, its power of p - 1 is: FieldElement{order: 11, num: 1}
-------------------------------
value of p is: 17
for element: FieldElement{order: 17, num: 1}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 2}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 3}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 4}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 5}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 6}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 7}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 8}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 9}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 10}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 11}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 12}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 13}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 14}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 15}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 16}, its power of p - 1 is: FieldElement{order: 17, num: 1}
-------------------------------
value of p is: 19
for element: FieldElement{order: 19, num: 1}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 2}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 3}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 4}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 5}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 6}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 7}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 8}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 9}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 10}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 11}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 12}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 13}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 14}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 15}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 16}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 17}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 18}, its power of p - 1 is: FieldElement{order: 19, num: 1}
-------------------------------
value of p is: 31
for element: FieldElement{order: 31, num: 1}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 2}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 3}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 4}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 5}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 6}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 7}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 8}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 9}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 10}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 11}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 12}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 13}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 14}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 15}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 16}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 17}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 18}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 19}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 20}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 21}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 22}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 23}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 24}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 25}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 26}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 27}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 28}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 29}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 30}, its power of p - 1 is: FieldElement{order: 31, num: 1}
-------------------------------

你可以看到集合中的所有元素都是1，无论字段的顺序如何，这意味着对于任何有限字段中的任意元素k和顺序p，我们会有：

k ^(p-1) % p == 1

这是一个重要结论，我们将在后续视频中使用它来驱动我们的加密算法。

有限域元素上最难的操作是除法，我们有乘法操作，对于字段中的元素3和7（顺序为19），它们的乘积是(3 * 7) % 19 = 2。现在给定两个字段元素2和7，我们如何得到7？我们定义一个除法操作，它是乘法的逆运算，即2 / 7 = 3，这相当直观。这里我们需要确保分母不是0。

记住在有限的定义中，如果a在字段中，那么还有一个b在字段中，使得a * b = 1。对于3 7 = 2（注意表示模顺序的乘法），如果我们能找到b，使得b * 7 = 1，那么我们就会有3 * 7 * b = 2 * b => 3 * (7 * b) = 2 * b => 3 = 2 * b，这意味着2 / 7是2乘以b的结果，b. 也就是说，如果我们想做除法a / b，我们可以找到b的乘法逆元，称之为c，并使用c与模顺序相乘。

现在问题来了，我们如何找到b的乘法逆元？记住我们上面的问题吗？b ^ (p - 1) % p = 1 => b * b ^(p-2) % p = 1 => b的乘法逆元是b ^ (p-2)。

如果你不能确定为什么对于给定元素b在字段中且b^(p-1) % p = 1，我们有一个小代码片段来获得结果，我们需要使其数学上稳固，然后我们就有了它的证明，结论b^(p-1) % p = 1被称为费马小定理：

对于任何字段元素k（k!=0）和顺序p，我们有{1, 2, 3 ..., p-1} <=> {k 1 % p, ..., k (p-1) %p} =>

[1 2 3... (p-1)] % p == (k1) (k2) ... (k* (p-1)) % p = k^(p-1) * [1 2 ... p-1] % p，两边消去[12...p-1]我们得到1 % p == k ^(p-1) % p => 1 == k^(p-1)%p

现在让我们看看如何使用代码实现除法操作：

go 复制代码 
func (f *FieldElement) Multiply(other *FieldElement) *FieldElement {
	f.checkOrder(other)
	// 模顺序进行乘法
	var op big.Int
	mul := op.Mul(f.num, other.num)
	return NewFieldElement(f.order, op.Mod(mul, f.order))
}

因为b ^ (p - 1) % p = 1，所以当我们计算字段元素k的T次方时，我们可以优化为首先获取t = T % (p-1)，然后计算k^(t) % p，这里是代码：

go 复制代码 
func (f *FieldElement) Power(power *big.Int) *FieldElement {
	/*
		k ^ (p-1) % p = 1，我们可以计算t = power % (p-1)
		然后k ^ power % p == k ^ t %p
	*/
	var op big.Int
	t := op.Mod(power, op.Sub(f.order, big.NewInt(int64(1))))
	powerRes := op.Exp(f.num, t, nil)
	modRes := op.Mod(powerRes, f.order)
	return NewFieldElement(f.order, modRes)
}

现在我们可以在main.go中检查我们的代码：

go 复制代码 
package main

import (
	ecc "elliptic_curve"
	"fmt"
	"math/big"
	"math/rand"
)

func main() {
	f2 := ecc.NewFieldElement(big.NewInt(int64(19)), big.NewInt(int64(2)))
	f7 := ecc.NewFieldElement(big.NewInt(int64(19)), big.NewInt(int64(7)))
	fmt.Printf("field element 2 / 7 with order 19 is %v\n", f2.Divide(f7))

	f46 := ecc.NewFieldElement(big.NewInt(57), big.NewInt(46))
	fmt.Printf("field element 46 * 46 with order 57: %v\n", f46.Multiply(f46))
	fmt.Printf("field element 46 ^ (58) is %v\n", f46.Power(big.NewInt(int64(58))))
}

运行上述代码我们得到以下结果：

``go

复制代码

field element 2 / 7 with order 19 is FieldElement{order: 19, num: 3}

field element 46 * 46 with order 57: FieldElement{order: 57, num: 7}

field element 46 ^ (58) is FieldElement{order: 57, num: 7}

这正是我们所期望的，这就是字段元素的实现。
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