Miller_Rabin 素性检验

Miler_Rabin

\(\forall x<p\) 存在：

• 费马小定理：\(x^{p-1}\equiv1\pmod p\)
• 二次探测定理：\(x^2\equiv 1\pmod p\Rightarrow x\equiv1\or x\equiv -1\pmod p\)

基底

可以参考 http://miller-rabin.appspot.com/。

• \(2,325,9375,28178,450775,9780504,1795265022\)：至少 \(2^{64}\)。

代码

inline int Miller_Rabin(i64 n) {
  const i64 P[] = { ..., 0 };
  if (n < 2) return 0;
  i64 m = n - 1, a, b; int k = 0;
  while (!(m & 1)) m >>= 1, ++k;
  for (int i = 0; P[i]; ++i) {
    if (!(a = P[i] % n)) return 1;
    a = b = pow(a, m, n);
    for (int j = 1; j <= k; ++j)
      if ((b = mul(a, a, n)) == 1 && a != 1 && a != n - 1) return 0;
    if (a != 1) return 0;
  }
  return 1;
}

i64 pow(i64 a, i64 k, i64 p) {
  i64 t = 1;
  for (; k; a = mul(a, a, p), k >>= 1) if (k & 1) t = mul(t, a, p);
  return t;
}
i64 mul(i64 a, i64 b, i64 p) {
  return ((a * b - (i64)((long double)a * b / p) * p) % p + p) % p;
}