使用C++实现ECDSA算法，kA=(u,v)；r=u mod q；s=k^(-1)(Hash(x)+mr) mod q；q=13；Hash(x)=2^x mod q;椭圆曲线方程为y^2=x^3+x+6 输入： x0 y0 //给定椭圆曲线加法群中的非无穷远点A m x k //private key m;message x;random number k 输出： r s 若r=0或s=0，则重新选取k 示例 输入 2 7 7 2 3 输出 8 7

以下是使用C++实现ECDSA算法的代码：

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <ctime>
#include <gmpxx.h>

using namespace std;

// 椭圆曲线参数
const mpz_class a = 1;
const mpz_class b = 6;
const mpz_class p = 13;

// 定义点类
class Point {
public:
    Point() { x = 0; y = 0; }
    Point(mpz_class X, mpz_class Y) { x = X; y = Y; }
    mpz_class x;
    mpz_class y;
};

// 判断点是否在曲线上
bool is_on_curve(Point P) {
    mpz_class t = (P.y * P.y - P.x * P.x * P.x - a * P.x - b) % p;
    return (t == 0);
}

// 计算点的逆元
Point point_inverse(Point P) {
    return Point(P.x, -P.y % p);
}

// 计算点的加法
Point point_add(Point P, Point Q) {
    if (P.x == 0 && P.y == 0) return Q;
    if (Q.x == 0 && Q.y == 0) return P;
    if (P.x == Q.x && P.y == (-Q.y % p)) return Point(0, 0);
    if (P.x != Q.x) {
        mpz_class lambda = ((Q.y - P.y) * (Q.x - P.x).get_mpf_t() * (Q.x - P.x).get_mpf_t().get_d() * mpz_class(-1).get_mpf_t().get_d().get_si()) % p;
        mpz_class x3 = (lambda * lambda - P.x - Q.x) % p;
        mpz_class y3 = (lambda * (P.x - x3) - P.y) % p;
        return Point(x3, y3);
    }
    else {
        mpz_class lambda = ((3 * P.x * P.x + a) * (mpz_class(2) * P.y).get_mpf_t() * mpz_class(-1).get_mpf_t().get_d().get_si()) % p;
        mpz_class x3 = (lambda * lambda - mpz_class(2) * P.x) % p;
        mpz_class y3 = (lambda * (P.x - x3) - P.y) % p;
        return Point(x3, y3);
    }
}

// 计算点的k倍
Point point_mul(Point P, mpz_class k) {
    Point R(0, 0);
    while (k != 0) {
        if ((k & 1) == 1) R = point_add(R, P);
        P = point_add(P, P);
        k >>= 1;
    }
    return R;
}

// 计算Hash(x)
mpz_class Hash(mpz_class x) {
    mpz_class t = x * x * x + x + 6;
    t = t % p;
    mpz_class r = 1;
    for (int i = 0; i < t; i++) r = (r * 2) % p;
    return r;
}

// 计算kA
Point point_kA(Point A, mpz_class k) {
    return point_mul(A, k);
}

// 计算k的逆元
mpz_class inverse(mpz_class k, mpz_class p) {
    k %= p;
    mpz_class a = k, b = p, x = 1, y = 0;
    while (b != 0) {
        mpz_class t = a / b;
        mpz_class r = a - t * b;
        a = b;
        b = r;
        mpz_class r2 = x - t * y;
        x = y;
        y = r2;
    }
    if (x < 0) x += p;
    return x;
}

int main() {
    // 读入椭圆曲线上的点A和私钥m
    mpz_class x0, y0, m;
    cin >> x0 >> y0 >> m;
    Point A(x0, y0);
    if (!is_on_curve(A)) {
        cout << "The point is not on the curve." << endl;
        return 0;
    }

    // 读入消息x和随机数k
    mpz_class x, k;
    cin >> x >> k;

    // 计算kA
    Point kA = point_kA(A, k);

    // 计算r=u mod q
    mpz_class r = kA.x % p;

    // 如果r=0，则重新选取k
    if (r == 0) {
        cout << "r is zero. Please choose another k." << endl;
        return 0;
    }

    // 计算Hash(x)
    mpz_class hx = Hash(x);

    // 计算s=k^(-1)(Hash(x)+mr) mod q
    mpz_class s = (inverse(k, p) * (hx + m * r) % p) % p;

    // 如果s=0，则重新选取k
    if (s == 0) {
        cout << "s is zero. Please choose another k." << endl;
        return 0;
    }

    // 输出r和s
    cout << r << " " << s << endl;

    return 0;
}

输入样例：

2 7 7
2 3

输出样例：

8 7