Prime Numbers

9.2 Enhancements to modular arithmetic 447
if(s ≥ N) s = s − N;
}
return s;
The binary divide and mul-mod algorithms, though illuminating, suffer
from a basic practical shortcoming: One is not taking due advantage of
multiple-bit arithmetic as is commonly available on any reasonably powerful
computer. One would like to perform multiple-bit operations within machine
registers, rather than just operating one bit at a time. For this reason, larger
bases than B = 2 are usually used, and many modern div implementations
invoke “Algorithm D,” see [Knuth 1981, p. 257], which is a finely tuned version
of the classical long division. That algorithm is a good example of one that
has more pseudocode complexity than does our binary Algorithm (9.1.3), yet
amounts to a great deal of optimization in actual programs.
9.2 Enhancements to modular arithmetic
The classical div and mod algorithms discussed in Section 9.1.3 all involve
some sort of explicit divide operation. For the binary algorithms given, this
division is trivial; that is, if 0 ≤ a