> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/advanced-math/fast-power.md).

# 快速幂与模运算

### 模运算性质

```
(a + b) % m = ((a % m) + (b % m)) % m
(a * b) % m = ((a % m) * (b % m)) % m
```

减法要加 m 防负：((a-b) % m + m) % m。**除法**不满足，需要逆元。

### 快速幂 O(log n)

求 a^b mod m，把指数按二进制拆分：

```cpp
long long qpow(long long a, long long b, long long m) {
    long long r = 1 % m;
    a %= m;
    while (b) {
        if (b & 1) r = r * a % m;
        a = a * a % m;
        b >>= 1;
    }
    return r;
}
```

### 原理

a^13 = a^(1101) = a^8 \* a^4 \* a^1，边平方边取用需要的幂。

### 快速乘（防溢出）

当 a\*b 超过 long long 时，用类似方法以加法代替乘法。

### 应用

* 大指数取模
* 求逆元：费马小定理 a^(m-2) mod m（m 为素数）
* 矩阵快速幂（递推加速）


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