Quick generalized power

Provided: ○ operation and the set as a group V configuration, a∈V, e is the identity element ○ operation.

That is, for any e satisfy a, there is e ○ a = a ○ e = a

We can remember

a ⁰ = e

aⁿ=a^n-1○a

It has the following properties

a^n+m=aⁿ○a^m

At this time, the calculation of a n-th power calculation on a fast power ○ can write

res=e;temp=a;

while(n)

{

        if(n&1)

                res=res○temp;

        temp=temp○temp;

        n>>=1;

}

return res;

Wherein the addition identity element is 0, the multiplication of 1

　　Because the same n + 0, n * 1 unchanged

 

Addition exponentiation

ll quick_mul(ll a, ll b, ll c)    
{
	ll res = 0;
	while (b != 0)
	{
		if (b % 2 == 1)
			res = (a + res) % c;
		a = (a + a) % c;
		b /= 2;
	}
	return res;
}

Multiply exponentiation

ll quick_pow(ll a, ll b, ll c)
{
    ll res = 1;
    while (b != 0)
    {
        if (b % 2 == 1)
            res=(res*a)%c    //优化   res = quick_mul(res, a, c);
        a = cq(a, a, c);
        b /= 2;
    }
    return res;
}

 

First, know (a * b)% c == (a% c) * (b% c) (a + b)% c == a% c + b% c

　　In particular to be understood as constantly first modulo c c Save, Save until enough

　　(a*b)%c==( ((a%c)*(c*x))  *  ((b%c)(c*y))  )%c               a+b)%c==( (a%c+a*x) +( b%c+b*y) )%c

　　Wherein the ((a% c) * (c * x)) represents a, x represents the unknown unrelated result, the (c * x) gave the same reason not Save

         

For a power of b

If b is an odd number a, b == a power of b / 2 * a power of b / 2 * a power

If b is an even number a, b == a power of b / 2 * a power of b / 2 power  

So the code above

 

Another point is that the power of multiplication operation can be optimized using addition exponentiation