添加miller rabin测试和pollard_pro方法

Author: zhitaogpt

math/数论算法模板总结.md

@@ -67,7 +67,79 @@ void segment_sieve(LL a,LL b)//[a,b]
     }
 }
 ```
+#大素数分解与大素数测试
 
+##miller_rabin
+
+已知最快的素数分解算法.$O(lgV)$
+
+```c++
+bool witness(LL a,LL n,LL u,LL t){
+	LL x0 = power_mod(a,u,n),x1;
+	for(int i=1 ;i<=t ; ++i){
+		x1 = mulmod(x0,x0,n);
+		if(x1==1 && x0!=1 && x0!=n-1)
+			return false;
+		x0 = x1;
+	}
+	if(x1 !=1)return false;
+	return true;
+}
+
+bool miller_rabin(LL n, int times = 20){
+	if(n<2)return false;
+	if(n==2)return true;
+	if(!(n&1))return false;
+	LL u = n-1,t =0;
+	while (u%2==0) {
+		t++;u>>=1;
+	}
+	while (times--) {
+		LL a = random(1,n-1);
+		//if(a == 0)std::cout << a << " "<<n<< " "<<u<<" " << t<<'\n';
+		if(!witness(a,n,u,t))return false;
+	}
+	return true;
+}
+```
+
+##pollard_rho
+分解一个合数$V$的运行时间$O(V^{1/4 })$
+```c++
+/*
+*pollard_rho分解n,
+*c : 随机迭代器，每次运行设置为随机种子往往更快.
+*/
+LL pollard_rho(LL n,LL c = 1){
+	LL x  = random(1,n);
+	LL i =1,k =2,y = x;
+	while (1) {
+		i++;
+		x = (mulmod(x,x,n)+c)%n;
+		LL d = gcd(y-x>=0?y-x:x-y,n);
+		if(d!=1 && d!=n)return d;//非平凡因子.
+		if(y==x)return n;//重复.
+		if(i==k){ y = x ; k<<=1;}//将x_1,2,4,8,16,..赋值为y.
+	}
+}
+```
+* 找出因子分解
+$O(V^{1/4}lgV)$
+```c++
+void find_factor(LL n,std::map<LL, int>  & m){
+	if(n<=1)return ;
+	if(miller_rabin(n)){
+		++m[n];
+		return ;
+	}
+	LL p = n;
+
+	while (p==n)p = pollard_rho(p,random(1,n));
+	find_factor(p,m);
+	find_factor(n/p,m);
+}
+
+```
 #euler phi函数
 $\phi(n) = n\prod_{i = 1}^{k}(1-\frac{1}{p_i})$
 证明详见《初等数论及其应用》