数论 欧几里德相关 模板

//非递归的gcd
int_t gcd(int_t a,int_t b){
while( b ){
int_t r = b;
b = a % b;
a = r;
}
return a;
}

//快速GCD
int Fastgcd(int a, int b)
{
if (a == 0) return b;
if (b == 0) return a;
if (!(a & 1) && !(b & 1))
return Fastgcd(a>>1, b>>1)<<1;
else if (!(b & 1))
return Fastgcd(a, b>>1);
else if (!(a & 1)) return Fastgcd(a>>1, b);
else return Fastgcd(abs(a - b), min(a, b));
}

//利用二进制计算a*b%mod
llt multiMod(llt a,llt b,llt mod){
llt ret = 0LL;
a %= mod;
while( b ){
if ( b & 1LL ) ret = ( ret + a ) % mod, --b;
b >>= 1LL;
a = ( a + a ) % mod;
}
return ret;
}

//计算a^b%mod
llt powerMod(llt a,llt b,llt mod){
llt ret = 1LL;
a %= mod;
while( b ){
if ( b & 1LL ) ret = multiMod(ret,a,mod),--b;
b >>= 1LL;
a = multiMod(a,a,mod);
}
return ret;
}

//非递归的扩展欧几里德算法
//返回a、b的gcd，同时x、y满足ax+by=gcd
int_t exEuclid(int_t a,int_t b,int_t&x,int_t&y){
int_t x0 = 1, y0 = 0;
int_t x1 = 0, y1 = 1;
x = 0; y = 1;
int_t r = a % b;
int_t q = ( a - r ) / b;
while( r ){
x = x0 - q * x1;
y = y0 - q * y1;
x0 = x1; y0 = y1;
x1 = x; y1 = y;
a = b; b = r; r = a % b;
q = ( a - r ) / b;
}
return b;
}

//求a相对于p的逆元，a、p互质才存在逆元
int_t inv(int_t a,int_t p){
int_t x,y;
int_t r = exEuclid(a,p,x,y);
if ( r != 1 ) return 0;
x = x % p;
if ( x < 0 ) x += p;
return x;
}