Modular exponentiation

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1. 递归实现：

unsigned long long modular_pow_recursive(unsigned long long base, int exponent, int modulus){    if (modulus == 1)    {           return 0;    }       if(exponent == 0)    {           return 1;    }           if(exponent & 1 == 1)    {           return base * modular_pow_recursive(base, exponent - 1, modulus) % modulus;    }       unsigned long long ret =  modular_pow_recursive(base, exponent >> 1,  modulus);    return ((ret * ret) % modulus;}

刚开始的时候，最后两行写成了如下的形式：

    int ret =  modular_pow_recursive(base, exponent >> 1,  modulus) %modulus;    return (ret * ret) % modulus

测试的时候发现结果不对，查了半天，才发现是有 overflow了。

改成前面代码里的形式，后者：

    int ret =  modular_pow_recursive(base, exponent >> 1,  modulus) %modulus;    return ((unsigned long long )ret * ret) % modulus

就可以了。

2. 迭代实现：

unsigned long long modular_pow(unsigned long long base, int exponent, int modulus){    if(modulus == 1)    {           return 0;    }       if(exponent == 0)    {           return 1;    }       unsigned long long  result = 1;    while(exponent > 0)    {           if( exponent & 1)        {            result = (result * base) % modulus;        }        base = (base * base) % modulus;        exponent = exponent >> 1;    }       return result;}

3. 一个比较容易想到的实现：

unsigned long long modular_pow_rude(unsigned long long base, int exponent, int modulus){    int i;    unsigned long long result = 1;    for(i = exponent; i > 0; i--)    {           result = (result * base) % modulus;    }           return result;}

测试代码：

int main(int argc, char **argv){    int base = 19;     int exponent = 78;    int modulus = 199879;    int ret = modular_pow_recursive(base, exponent, modulus);    ret = modular_pow(base, exponent, modulus);    return ret;}

References:
https://en.wikipedia.org/wiki/Modular_exponentiation

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