By Henry Ibstedt

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Since b ≥ 3 and p < bk , we have 3 p < bk+1 . Thus 0 ≤ z −q p < bk+1 , and so r = z −q p after step 3. Hence, at most two subtractions at step 4 are required to obtain 0 ≤ r < p, and then r = z mod p. 14) (i) A natural choice for the base is b = 2 L where L is near the word size of the processor. (ii) Other than the calculation of µ (which is done once per modulus), the divisions required are simple shifts of the base-b representation. (iii) Let z = z/bk−1 . Note that z and µ have at most k + 1 base-b digits.

3. Depth-2 splits for 192-bit integers. The product x y using (a) has three 96×96 multiplications. Each is performed with a 32×32 and two 64×64 (each requiring three 32×32) multiplications, for a total of 21 multiplications of size 32 ×32. Using (b) or (c), only 18 multiplications of size 32×32 are required. As a second illustration, consider Karatsuba-Ofman applied to 192-bit integers, again with W = 32. 3. 3(a) will require 21, while (b) and (c) use 18. The basic idea is that multiplication of 3l-bit integers x = x2 22l + x1 2l + x0 and y = y2 22l + y1 2l + y0 can be done as x y = (x2 22l + x1 2l + x0 ) · (y2 22l + y1 2l + y0 ) = x2 y2 24l + (x2 y1 + x1 y2 )23l + (x2 y0 + x0 y2 + x1 y1 )22l + (x1 y0 + x0 y1 )2l + x0 y0 = x2 · y2 24l + [(x2 + x1 ) · (y2 + y1 ) − x2 y2 − x1 · y1 ]23l + [(x2 + x0 ) · (y2 + y0 ) − x2 y2 − x0 · y0 + x1 y1 ]22l + [(x1 + x0 ) · (y1 + y0 ) − x1 y1 − x0 y0 ]2l + x0 y0 for a total of six multiplications of l-bit integers.

The reduction step can be accelerated considerably when the modulus p has a special form. 6. The algorithms presented here are well suited for software implementation. We assume that the implementation platform has a W -bit architecture where W is a multiple of 8. Workstations are commonly 64- or 32-bit architectures. Low-power or inexpensive components may have smaller W , for example, some embedded systems are 16-bit and smartcards may have W = 8. The bits of a W -bit word U are numbered from 0 to W − 1, with the rightmost bit of U designated as bit 0.