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:

    “With these I would be.
    And with water: the waves coming forward, without cessation,
    The waves, altered by sand-bars, beds of kelp, miscellaneous
    driftwood,
    Topped by cross-winds, tugged at by sinuous undercurrents
    The tide rustling in, sliding between the ridges of stone,
    The tongues of water, creeping in, quietly.”
    —Theodore Roethke (1908–1963)

    “As the end of the century approaches, all our culture is like the culture of flies at the beginning of winter. Having lost their agility, dreamy and demented, they turn slowly about the window in the first icy mists of morning. They give themselves a last wash and brush-up, their ocellated eyes roll, and they fall down the curtains.”
    —Jean Baudrillard (b. 1929)

    “There is no method but to be very intelligent.”
    —T.S. (Thomas Stearns)