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golang 實(shí)現(xiàn)比特幣內(nèi)核之處理橢圓曲線中的天文數(shù)字

 更新時(shí)間：2024年11月06日 14:47:55   作者：coding 迪斯尼  
比特幣密碼學(xué)中涉及到的大數(shù)運(yùn)算超出常規(guī)整數(shù)范圍,需使用golang的big包進(jìn)行處理,通過(guò)使用big.Int類型,能有效避免整數(shù)溢出,并保持邏輯正確性,測(cè)試展示了在不同質(zhì)數(shù)模下的運(yùn)算結(jié)果,驗(yàn)證了邏輯的準(zhǔn)確性,此外,探討了費(fèi)馬小定理在有限字段除法運(yùn)算中的應(yīng)用

在比特幣密碼學(xué)中，我們需要處理天文數(shù)字，這個(gè)數(shù)字是如此巨大，以至于它很容易超出我們宇宙中原子的總數(shù)，也許 64 位的值不足以表示這個(gè)數(shù)字，而像加、乘、冪這樣的操作如果使用 64 位整數(shù)會(huì)導(dǎo)致溢出，因此我們可能需要借助 golang 的 big 包，我們將通過(guò)使用 big.Int 來(lái)表示其值字段來(lái)更改 FieldNumber 的代碼，代碼將如下所示：

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)
}

現(xiàn)在我們需要確保這些更改不會(huì)破壞我們的邏輯，讓我們?cè)俅芜\(yùn)行測(cè)試，在 main.go 中，我們有以下代碼：

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()
}

運(yùn)行上述代碼將獲得以下結(jié)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}

通過(guò)檢查結(jié)果，我們可以確保 FieldElement 中的更改不會(huì)破壞我們之前的邏輯?，F(xiàn)在讓我們考慮以下問(wèn)題：
p = 7, 11, 17, 19, 31，以下集合會(huì)是什么：
{1 ^(p-1), 2 ^ (p-1), … (p-1)^(p-1)}
讓我們?cè)?main.go 中編寫(xiě)代碼來(lái)解決它：

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()
}

結(jié)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，無(wú)論字段的順序如何，這意味著對(duì)于任何有限字段中的任意元素k和順序p，我們會(huì)有：
k ^(p-1) % p == 1
這是一個(gè)重要結(jié)論，我們將在后續(xù)視頻中使用它來(lái)驅(qū)動(dòng)我們的加密算法。

有限域元素上最難的操作是除法，我們有乘法操作，對(duì)于字段中的元素3和7（順序?yàn)?9），它們的乘積是(3 * 7) % 19 = 2?，F(xiàn)在給定兩個(gè)字段元素2和7，我們?nèi)绾蔚玫?？我們定義一個(gè)除法操作，它是乘法的逆運(yùn)算，即2 / 7 = 3，這相當(dāng)直觀。這里我們需要確保分母不是0。

記住在有限的定義中，如果a在字段中，那么還有一個(gè)b在字段中，使得a * b = 1。對(duì)于3 7 = 2（注意表示模順序的乘法），如果我們能找到b，使得b * 7 = 1，那么我們就會(huì)有3 * 7 * b = 2 * b => 3 * (7 * b) = 2 * b => 3 = 2 * b，這意味著2 / 7是2乘以b的結(jié)果，b. 也就是說(shuō)，如果我們想做除法a / b，我們可以找到b的乘法逆元，稱之為c，并使用c與模順序相乘。

現(xiàn)在問(wèn)題來(lái)了，我們?nèi)绾握业絙的乘法逆元？記住我們上面的問(wèn)題嗎？b ^ (p - 1) % p = 1 => b * b ^(p-2) % p = 1 => b的乘法逆元是b ^ (p-2)。

如果你不能確定為什么對(duì)于給定元素b在字段中且b^(p-1) % p = 1，我們有一個(gè)小代碼片段來(lái)獲得結(jié)果，我們需要使其數(shù)學(xué)上穩(wěn)固，然后我們就有了它的證明，結(jié)論b^(p-1) % p = 1被稱為費(fèi)馬小定理：

對(duì)于任何字段元素k（k!=0）和順序p，我們有{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，兩邊消去[12…p-1]我們得到1 % p == k ^(p-1) % p => 1 == k^(p-1)%p

現(xiàn)在讓我們看看如何使用代碼實(shí)現(xiàn)除法操作：

func (f *FieldElement) Multiply(other *FieldElement) *FieldElement {
	f.checkOrder(other)
	// 模順序進(jìn)行乘法
	var op big.Int
	mul := op.Mul(f.num, other.num)
	return NewFieldElement(f.order, op.Mod(mul, f.order))
}

因?yàn)閎 ^ (p - 1) % p = 1，所以當(dāng)我們計(jì)算字段元素k的T次方時(shí)，我們可以優(yōu)化為首先獲取t = T % (p-1)，然后計(jì)算k^(t) % p，這里是代碼：

func (f *FieldElement) Power(power *big.Int) *FieldElement {
	/*
		k ^ (p-1) % p = 1，我們可以計(jì)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)
}

現(xiàn)在我們可以在main.go中檢查我們的代碼：

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))))
}

運(yùn)行上述代碼我們得到以下結(jié)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}

這正是我們所期望的，這就是字段元素的實(shí)現(xiàn)。

到此這篇關(guān)于golang 實(shí)現(xiàn)比特幣內(nèi)核:處理橢圓曲線中的天文數(shù)字的文章就介紹到這了,更多相關(guān)golang比特幣內(nèi)核內(nèi)容請(qǐng)搜索腳本之家以前的文章或繼續(xù)瀏覽下面的相關(guān)文章希望大家以后多多支持腳本之家！

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