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:

    “Argument is conclusive ... but ... it does not remove doubt, so that the mind may rest in the sure knowledge of the truth, unless it finds it by the method of experiment.... For if any man who never saw fire proved by satisfactory arguments that fire burns ... his hearer’s mind would never be satisfied, nor would he avoid the fire until he put his hand in it ... that he might learn by experiment what argument taught.”
    —Roger Bacon (c. 1214–1294)