Prime Numbers

Chapter 9
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 multipleprecision
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.
9.1 Tour of “grammar-school” methods
9.1.1 Multiplication
One of the most common technical aspects of our culture is the classical,
or shall we say “grammar-school,” method of long multiplication. Though we
shall eventually concentrate on fast, modern methods of remarkable efficiency,
the grammar-school multiply remains important, especially when the relevant
integers are not too large, and itself allows some speed enhancements. In the
typical manifestation of the algorithm, one simply writes out, one below the
other, the two integers to be multiplied, then constructs a parallelogram of
digitwise products. Actually, the parallelogram is a rhombus, and to complete
the multiply we need only add up the columns of the rhombus, with carry. If
each of x, y to be multiplied has D digits in some given base B (also called
the “radix”), then the total number of operations required to calculate xy is
O(D 2 ), because that is how many entries appear in the rhombus. Here, an
“operation” is either a multiply or an add of two numbers each of size B. We
shall refer to such a fundamental, digitwise, multiply as a “size-B multiply.”
A formal exposition of grammar-school multiply is simple but illuminating,
especially in view of later enhancements. We start with two definitions:
Definition 9.1.1. The base-B representation of a nonnegative integer x
is the shortest sequence of integer digits (xi) such that each digit satisfies