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Exponentiation by squaring facts for kids

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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 xn for a positive integer n where n > 0:

Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 >\mbox{2 is odd} \end{cases}

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.

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