Repeated squaring

Repeated squaring, or repeated doubling is an algorithm that computes integer powers of a number quickly. The general problem is to compute $$x^y$$ for an arbitrary integer y. The naive method, doing y multiplications of x, is very slow. It can be sped up by repeatedly squaring x until the current power of x exceeds y, and then collecting the "useful" powers.

Example
So, which powers of x are useful? Consider the concrete example of computing $$3^{13}$$. The binary representation of 13 is $${(1101)}_2$$, so $$8 + 4 + 1 = 13$$.

Notice that each result is the square of the previous result, and hence can be computed in one multiplication.

$$ 3^1 \times 3^4 \times 3^8 = 3^{13} = 1594323$$

Pseudo-code
// Calculates n to the p power, where p is a positive number. func power( var n as integer, var p as integer ) if p = 0 return 1 if p = 1 return n  if p is odd return n * power( n * n, (p-1) / 2 ) else return power( n * n, p / 2 ) end func It is not neccesary to keep all the powers of x in memory, only a product accumulator and the last power of x is neccesary.

Note that the algorithm does only $$O(\lg y)$$ multiplications, since it has to do no more than twice as many multiplcations as the number of bits representing y.

In general, we use the equation $$x^y = (x^{y~{\rm div}~2})^2 \cdot x^{y\bmod 2}$$.