Chapter 10: Basic Number-Theoretic Functions

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Overview

I am dying to hear about it, since I always thought number theory was the Queen of Mathematics—the purest branch of mathematics—the one branch of mathematics which has NO applications!

—D. R. Hofstadter, Gödel, Escher, Bach

NOW THAT WE ARE FITTED out with a sturdy tool box of arithmetic functions that we developed in the previous chapters, we turn our attention to the implementation of several fundamental algorithms from the realm of number theory. The number-theoretic functions discussed in the following chapters form a collection that on the one hand exemplifies the application of the arithmetic of large numbers and on the other forms a useful foundation for more complex number-theoretic calculations and cryptographic applications. The resources provided here can be extended in a number of directions, so that for almost every type of application the necessary tools can be assembled with the demonstrated methods.

The algorithms on which the following implementations are based are drawn primarily from the publications [Cohe], [HKW], [Knut], [Kran], and [Rose], where as previously, we have placed particular value on efficiency and on as broad a range of application as possible.

The following sections contain the minimum of mathematical theory required to explicate the functions that we present and their possibilities for application. We would like, after all, to have some benefit from all the effort that will be required in dealing with this material. Those readers who are interested in a more thoroughgoing introduction to number theory are referred to the books [Bund] and [Rose]. In [Cohe] in particular the algorithmic aspects of number theory are considered and are treated clearly and concisely. An informative overview of applications of number theory is offered by [Schr], while cryptographic aspects of number theory are treated in [Kobl].

In this chapter we shall be concerned with, among other things, the calculation of the greatest common divisor and the least common multiple of large numbers, the multiplicative properties of residue class rings, the identification of quadratic residues and the calculation of square roots in residue class rings, the Chinese remainder theorem for solving systems of linear congruences, and the identification of prime numbers. We shall supplement the theoretical foundations of these topics with practical tips and explanations, and we shall develop several functions that embody a realization of the algorithms that we describe and make them usable in many practical applications.

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