golang 实现比特币内核之处理椭圆曲线中的天文数字

更新时间：2024年11月06日 14:47:55 作者：coding 迪斯尼

比特币密码学中涉及到的大数运算超出常规整数范围,需使用golang的big包进行处理,通过使用big.Int类型,能有效避免整数溢出,并保持逻辑正确性,测试展示了在不同质数模下的运算结果,验证了逻辑的准确性,此外,探讨了费马小定理在有限字段除法运算中的应用

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：

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：

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：

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：

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：

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：

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：

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：

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：

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。

到此这篇关于golang 实现比特币内核:处理椭圆曲线中的天文数字的文章就介绍到这了,更多相关golang比特币内核内容请搜索脚本之家以前的文章或继续浏览下面的相关文章希望大家以后多多支持脚本之家！

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golang 实现比特币内核之处理椭圆曲线中的天文数字

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