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)

    “If a man was tossed out of a window when an infant, and so made a cripple for life, or scared out of his wits by the Indians, it is regretted chiefly because he was thus incapacitated for—business! I think that there is nothing, not even crime, more opposed to poetry, to philosophy, ay, to life itself, than this incessant business.”
    —Henry David Thoreau (1817–1862)

    “Women stand related to beautiful nature around us, and the enamoured youth mixes their form with moon and stars, with woods and waters, and the pomp of summer. They heal us of awkwardness by their words and looks. We observe their intellectual influence on the most serious student. They refine and clear his mind: teach him to put a pleasing method into what is dry and difficult.”
    —Ralph Waldo Emerson (1803–1882)