欧拉定理与扩展欧拉定理学习笔记

关于欧拉函数及其求法的复习请戳这里

这里是欧拉定理及其扩展的证明

一句话总结:

\(a^b​\) \(mod​\) \(m​\) \(≡​\) \(a^b​\) \((b < phi(m))​\)

\(a^b\) \(mod\) \(m\) \(≡\) \(a^{b \% phi(m)+phi(m)}\) \((b >= phi(m))\)

实际上不需要记欧拉定理,后面这个在应用上已经可以涵盖前面的了。

用于优化指数非常大的情况。

#include <bits/stdc++.h>
using namespace std;

#define LL long long

const int N = 1000010;

int tot, divv[N];
int a, b, m;

int fpow (int a, int b, int m) {
	int res = 1;
	while (b) {
		if (b & 1) {
			res = 1LL * res * a % m;
		}
		a = 1LL * a * a % m;
		b >>= 1;
	}
	return res;
}

int main () {
	cin >> a >> m;
	int mx = sqrt (m), tmp = m;
	for (int i = 2; i <= mx; ++i) {
		if (tmp % i == 0) {
			divv[++tot] = i;
			while (tmp % i == 0) tmp /= i;
		}
	}
	if (tmp != 1) divv[++tot] = tmp;
	int _phi = m;
	for (int i = 1; i <= tot; ++i) {
		_phi = _phi / divv[i] * (divv[i] - 1);
	}
	int b = 0, ch = getchar (), type = 1;
	while ('9' < ch || ch < '0') ch = getchar ();
	while ('0' <= ch && ch <= '9') {
		b = b * 10 + ch - '0';
		if (b >= _phi) {
			type = 2;
			b = b % _phi;
		}
		ch = getchar ();
	}
	if (type == 2) b = b % _phi + _phi;
	cout << fpow (a, b, m);
} 

posted @ 2019-03-07 16:50  maomao9173  阅读(234)  评论(0编辑  收藏  举报