# 二次剩余方程求解

((b+sqrt(w))^p=b^p+w^(p/2)(modp)=b+w^((p-1)/2)*sqrt(w)(modp)=b-sqrt(w)(modp)

x^2=((b+sqrt(w))^(p+1)=(b-sqrt(w))(b+sqrt(w))=b^2-w=a(modp),得证。

`#include <iostream>using namespace std;#define LL __int64 LL w;struct Point//x + y*sqrt(w){LL x;LL y;};LL mod(LL a, LL p){a %= p;if (a < 0){a += p;}return a;}Point point_mul(Point a, Point b, LL p){Point res;res.x = mod(a.x * b.x, p);res.x += mod(w * mod(a.y * b.y, p), p);res.x = mod(res.x, p);res.y = mod(a.x * b.y, p);res.y += mod(a.y * b.x, p);res.y = mod(res.y , p);return res;}Point power(Point a, LL b, LL p){Point res;res.x = 1;res.y = 0;while(b){if (b & 1){res = point_mul(res, a, p);}a = point_mul(a, a, p);b = b >> 1;}return res;}LL quick_power(LL a, LL b, LL p)//(a^b)%p{LL res = 1;while(b){if (b & 1){res = (res * a) % p;}a = (a * a) % p;b = b >> 1;}return res;}LL Legendre(LL a, LL p) // a^((p-1)/2){return quick_power(a, (p - 1) >> 1, p);}LL equation_solve(LL b, LL p)//求解x^2=b(%p)方程解{if ((Legendre(b, p) + 1) % p == 0){return -1;//表示没有解}LL a, t;while(true){a = rand() % p;t = a * a - b;t = mod(t, p);if ((Legendre(t, p) + 1) % p == 0){break;}}w = t;Point temp, res;temp.x = a;temp.y = 1;res = power(temp, (p + 1) >> 1, p);return res.x;}int main(){LL b, p;scanf("%I64d %I64d", &b, &p);printf("%I64d\n", equation_solve(b, p));//输出为-1表示，不存在解return 0;}`

0 0