Miller_Rabin、 Pollard_rho Template

Multiply and pow Function:

//计算 (a*b)%c.   a,b都是ll的数，直接相乘可能溢出的

//  a,b,c <2^63

ll mult_modq(ll a,ll b,ll c){

    a %= c;

    b %= c;

    ll ret = 0;

    while(b){

        if(b & 1){ret += a;ret %= c;}

        a <<= 1;

        if(a >= c)a %= c;

        b >>= 1;

    }

    return ret;

}



//计算  x^n %c

ll pow_mod(ll x,ll n,ll mod){

    if(n == 1)return x%mod;

    x %= mod;

    ll tmp = x;

    ll ret = 1;

    while(n){

        if(n & 1) ret = mult_mod(ret, tmp, mod);

        tmp = mult_mod(tmp, tmp, mod);

        n >>= 1;

    }

    return ret;

}

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Miller_Rabin Prime Number test：

return TRUE when Prime Number (BE POSSIBLITY)

return FALSE when not a Prime Number

//以a为基,n-1=x*2^t      a^(n-1)=1(mod n)  验证n是不是合数

//一定是合数返回true,不一定返回false

bool check(ll a,ll n,ll x,ll t){

    ll ret = pow_mod(a, x, n);

    ll last = ret;

    for(int i = 1; i <= t; ++i){

        ret = mult_mod(ret,ret,n);

        if(ret == 1 && last !=1 && last != n - 1) return true;//合数

        last = ret;

    }

    if(ret != 1) return true;

    return false;

}



// Miller_Rabin()算法素数判定

//是素数返回true.(可能是伪素数，但概率极小)

//合数返回false;



bool Miller_Rabin(ll n){

    if(n < 2)   return false;

    if(n == 2)    return true;

    if((n & 1) == 0)    return false;//偶数

    ll x = n - 1;

    ll t = 0;

    while((x & 1) == 0){x >>= 1; ++t;}

    for(int i = 0; i <S ; ++i){

        ll a = rand() % (n - 1) + 1;

        if(check(a,n,x,t))

            return false;//合数

    }

    return true;

}

 

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Pollard_rho Algorithm

The quality factor decomposition :

ll factor[100];//质因数分解结果（刚返回时是无序的）

int tol;//质因数的个数。数组小标从0开始



ll gcd(ll a,ll b){

    if(a == 0)  return 1;

    if(a < 0) return gcd(-a,b);

    while(b){

        ll t = a % b;

        a = b;

        b = t;

    }

    return a;

}



ll Pollard_rho(ll x,ll c){

    ll i = 1, k = 2;

    ll x0 = rand()%x;

    ll y = x0;

    while(1){

        ++i;

        x0 = (mult_mod(x0,x0,x)+c)%x;

        ll d = gcd(y-x0,x);

        if(d != 1 && d != x) return d;

        if(y == x0) return x;

        if(i == k){y = x0; k += k;}

    }

}

//对n进行素因子分解

void findfac(ll n){

    if(Miller_Rabin(n)){

        factor[tol++] = n;

        return;

    }

    ll p = n;

    while(p >= n)   p = Pollard_rho(p,rand()%(n-1)+1);

    findfac(p);

    findfac(n/p);

}