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# Exponentiation by squaring

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**Exponentiating by squaring** is an algorithm. It is used for quickly working out large integer powers of a number. It is also known as the **square-and-multiply** algorithm or **binary exponentiation**. It uses the binary expansion of the exponent. It is of quite general use, for example in modular arithmetic.

The algorithm is believed to have first been documented in the Sanskrit book *Chandah-sûtra*, about 200 BC.

## Squaring algorithm

The following recursive algorithm computes *x*^{n} for a positive integer *n* where n > 0:

- [math] \mbox{Power}(x,\,n)= \begin{cases} x, & \mbox{if }n\mbox{ = 1} \\ \mbox{Power}(x^2,\,n/2), & \mbox{if }n\mbox{ is even} \\ x\times\mbox{Power}(x^2,\,(n-1)/2), & \mbox{if }n \gt \mbox{2 is odd} \end{cases} [/math]

This algorithm is much faster than the ordinary method to compute such a value. Multipliying x by itself, n operations are needed to calculate *x ^{n}*. With the method shown above, only log2(n) operations are needed.