Lectures notes on number theory for computer science

Abstract

This brief, in the form of an e-book, is a collection of notes that cover elementary and medium level number theory with a target audience of primarily computer science students. It can be used in the number theory portion of a discrete mathematics course, or a course on the mathematical foundations of computer science, or as background material for a cryptography course. Thematically it is split into five areas that map to chapters. The first chapter is introductory and covers topics including divisibility, prime numbers, and modular arithmetic including modular linear equations. The second chapter covers additional topics such as Euler's totient function, units and inverses, the Chinese remainder theorem, and Fermat's and Euler's theorems. The following chapter covers primitive roots, quadratic residues, the Jacobi and Legendre symbols, Gauss's lemma and Eisenstein's theorem, and briefly discusses applications of number theory to cryptography. The fourth chapter is focused on traditional primality testing methods covering Miller's algorithms, Rabin's conversion of a Miller algorithm into a probabilistic primality test algorithm, Solovay-Strassen's algorithm and several other peripheral results including Carmichael numbers and the equivalence of Miller's two algorithms. Finally the last brief chapter can be viewed as an introduction to more advanced elements of number theory and its coverage includes multiplicative functions, the M\"obius function, Dirichlet products and Dirichlet and M\"obius inversions. Different parts of this e-book are for freshman to senior undergraduate students in computing and in particular computer science. Graduate students with limited exposure to number theory can use it to acquire a background suitable for typical cryptography courses at the master's level.