Prime Numbers

446 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
quotient is guaranteed to be in the interval [1,B − 1]. The quotient here is,
of course, the leading base-B digit of the final div result. One then replaces x
with x − m⌊x/m⌋, and divides m by B, that is, shifts m down by one digit,
and so on recursively. This sketch shows us right off that for certain bases
B, things are relatively simple. In fact, if one adopts binary representations
(B = 2), then a complete div algorithm can be effected such that there are
no multiplies at all. The method can actually be of practical interest, especially
on machinery that has addition, subtraction, bit-shifting (left-shift
means multiply-by-2, right-shift means divide-by-2), but little else in the way
of operations. Explicitly, we proceed as follows:
Algorithm 9.1.3 (Classical binary divide). Given positive integers x ≥ N,
this algorithm performs the div operation, returning ⌊x/N⌋. (See Exercise 9.7 for
the matter of also returning the value x mod N.)
1. [Initialize]
Find the unique integer b such that 2 b N ≤ x 0 being the high bit.
1. [Initialize]
s =0;
2. [Loop over D bits]
for(D − 1 ≥ j ≥ 0) {
s =2s;
if(s ≥ N) s = s − N;
if(xj == 1) s = s + y;