Exponentiation By Squaring - Sliding Window Method

Sliding Window Method

This method is an efficient variant of the 2k-ary method. For example, to calculate the exponent 398 which has binary expansion (110 001 110)2, we take a window of length 3 using the 2k-ary method algorithm we calculate 1,x3,x6,x12,x24,x48,x49,x98,x99,x198,x199,x398. But, we can also compute 1,x3,x6,x12,x24,x48,x96,x192,x199, x398 which saves one multiplication and amounts to evaluating (110 001 110)n2

Here is the general algorithm:

Algorithm:

Input
An element 'x' of 'G',a non negative integer n=(nl,nl-1,...,n0)2, a parameter k>0 and the pre-computed values x3,x5,....
Output
The element xn in G

Algorithm:

1. y := 1 and i := l-1 2. while i > -1 do 3. if ni=0 then y:=y2 and i:=i-1 4. else 5. s:=max{i-k+1,0} 6. while ns=0 do s:=s+1 7. for h:=1 to i-s+1 do y:=y2 8. u:=(ni,ni-1,....,ns)2 9. y:=y*xu 10. i:=s-1 11. return y

Note:

  1. In line 6 the loop finds the longest string of length less than or equal to 'k' which ends in a non zero value. Also not all odd powers of 2 up to need be computed and only those specifically involved in the computation need be considered.

Read more about this topic:  Exponentiation By Squaring

Famous quotes containing the words sliding, window and/or method:

    “Here at the fountain’s sliding foot,
    Or at some fruit-tree’s mossy root,
    Casting the body’s vest aside,
    My soul into the boughs does glide:
    There, like a bird, it sits and sings,
    Then whets and combs its silver wings,
    And, till prepared for longer flights,
    Waves in its plumes the various light.”
    —Andrew Marvell (1621–1678)

    “Our memory is like a shop in the window of which is exposed now one, now another photograph of the same person. And as a rule the most recent exhibit remains for some time the only one to be seen.”
    —Marcel Proust (1871–1922)

    “I am not afraid of the priests in the long-run. Scientific method is the white ant which will slowly but surely destroy their fortifications. And the importance of scientific method in modern practical life—always growing and increasing—is the guarantee for the gradual emancipation of the ignorant upper and lower classes, the former of whom especially are the strength of the priests.”
    —Thomas Henry Huxley (1825–95)