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;
}