Fast Algorithms for Large-Integer Arithmetic

In this chapter we explore the galaxy of “fast” algorithms that admit of applications in prime number and factorization computations. In modern times, it is of paramount importance to be able to manipulate multiple-precision integers, meaning integers that in practice, on prevailing machinery, have to be broken up into pieces, with machine operations to involve those pieces, with a view to eventual reassembly of desired results. Although multiple-precision addition and subtraction of, integers is quite common in numerical studies, we assume that notions of these very simple fundamental operations are understood, and start with multiplication, which is perhaps the simplest arithmetic algorithm whose classical form admits of genuine enhancements.