Fast exponentiation

Updated: 15.09.2019

Fast exponentiation is an algorithm that allows you to raise any number to a natural power for a reduced number of multiplications.

Description of the algorithm

For any number x and even degree n , the identity holds:

x n = (x n / 2 ) 2 = x n / 2 ⋅ x n / 2

This is the basis of the rapid exponentiation algorithm. Since such a partition allows, for one operation of multiplication, to halve the calculated degree.

For the case of odd degree, it is enough to lower it by one:

x n = x n - 1 ⋅ x , while (n - 1) is an even number.

Recursive implementation of rapid exponentiation

static long Power(long x, int n)
{
    if (n == 0)
    {
        return 1;
    }

    if (n% 2 == 0)
    {
        var p = Power(x, n / 2);
        return p * p;
    }
    else
    {
        return x * Power(x, n - 1);
    }
}

For optimization, you can replace the parity and division by 2 bit operations:

static long Power(long x, int n)
{
    if (n == 0)
    {
        return 1;
    }
    
    // for an even number, the last bit is zero
    if ((n & 1) == 0)
    {
        // shifting one bit to the right is equivalent to dividing by two
        var p = Power(x, n >> 1);
        return p * p;
    }
    else
    {
        return x * Power(x, n - 1);
    }
}

Iterative implementation

In this method of rapid exponentiation, we also use the optimization of parity and division by two:

static long Power(long x, int n)
{
    var result = 1L;
    while (n> 0)
    {
        if ((n & 1) == 0)
        {
            x *= x;
            n >>= 1;
        }
        else
        {
            result *= x;
            --n;
        }
    }

    return result;
}
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