Pollard_rho 因数分解

Int64以内Rabin-Miller强伪素数测试和Pollard 因数分解的算法实现

选取随机数\(ａ\) 随机数\(ｂ\)，检查\(gcd(a - b, n)\)是否大于１，若大于１则\(a - b\)是\(ｎ\)的一个因数

实现１：floyd判环

利用多项式\(f(x)\)迭代出\({x_0, x_1, \dots, x_k}\)

设定\(x = y = x_0\)的初始值，选用多项式进行迭代，每次：\(x = f(x)\), \(y = f(f(y))\)，即：\(x = x_k, y = x_{2k}\)当\(x == y\)时出现循环

设\(x = y = 2\)，\(f(n) = n^2 + a\)

typedef long long ll;

ll mul_mod(ll a, ll b, ll m){

    ll ans = 0, exp = a;

    while(a >= m) a -= m;

    while(b){

        if(b & 1){

            ans += exp;

            while(ans >= m) ans -= m;

        }

        exp += exp;

        while(exp >= m) exp -= m;

        b >>= 1;

    }

    return ans;

}

ll pollard_rho(ll n, int a){

    ll x = 2, y = 2, d = 1;

    while(d == 1){

        x = mul_mod(x, x, n) + a;

        y = mul_mod(y, y, n) + a;

        y = mul_mod(y, y, n) + a;

        d = __gcd((x >= y ? x - y : y - x), n);

    }

    if(d == n) return pollard_rho(n, a + 1);

    return d;

}

实现２: brent判环（更高效）

不同于floyd每次计算\(x_k, x_{2k}\)进行判断,brent每次只计算\(x_k\),当ｋ是2的方幂时,\(y = x_k\),每次计算\(d = gcd(x_k - y, n)\)

typedef long long ll;

ll mul_mod(ll a, ll b, ll m){

    ll ans = 0, exp = a;

    while(a >= m) a -= m;

    while(b){

        if(b & 1){

            ans += exp;

            while(ans >= m) ans -= m;

        }

        exp += exp;

        while(exp >= m) exp -= m;

        b >>= 1;

    }

    return ans;

}

ll pollard_rho(ll n, int a){

    ll x = 2, y = 2, d = 1, k = 0, i = 1;

    while(d == 1){

        ++k;

        x = mul_mod(x, x, n) + a;

        d = __gcd(x >= y ? x - y : y - x, n);

        if(k == i){

            y = x;

            i <<= 1;

        }

    }

    if(d == n) return pollard_rho(n, a + 1);

    return d;

}