Exponentiation By Squaring - 2k-ary Method

2k-ary Method

This algorithm calculates the value of xn after expanding the exponent in base 2k. It was first proposed by Brauer in 1939. In the algorithm below we make use of the following function f(0) = (k,0) and f(m) = (s,u) where m = u·2s with u odd.

Algorithm:

Input
An element x of G, a parameter k > 0, a non-negative integer n = (nl−1, nl−2, ..., n0)2k and the precomputed values x3, x5, ..., .
Output
The element xn in G
1. y := 1 and i := l-1 2. While i>=0 do 3. (s,u) := f(ni) 4. for j:=1 to k-s do 5. y := y2 6. y := y*xu 7. for j:=1 to s do 8. y := y2 9. i := i-1 10. return y

For optimal efficiency, k should be the smallest integer satisfying

Read more about this topic:  Exponentiation By Squaring

Famous quotes containing the word method:

    “The good husband finds method as efficient in the packing of fire-wood in a shed, or in the harvesting of fruits in the cellar, as in Peninsular campaigns or the files of the Department of State.”
    —Ralph Waldo Emerson (1803–1882)