Primality Testing and Integer Factorization in Public-Key Cryptography

Primality testing and integer factorization, as identified by Gauss in his "Disquisitiones Arithmeticae", Article 329, in 1801, are the two most fundamental problems (as well as the two most important research fields) in computational number theory. With the advent of modern computers, unexpected applications have also been found in primality testing and integer factorization. Primality Testing and Integer Factorization in Public-Key Cryptography introduces various algorithms for primality testing and integer factorization, with their applications in public-key cryptography and information security. More specifically, this book explores basic concepts and results in number theory in Chapter 1. Chapter 2 discusses various algorithms for primality testing and prime number generation, with an emphasis on the Miller-Rabin probabilistic test, the Goldwasser-Kilian and Atkin-Morain elliptic curve tests, and the Agrawal-Kayal-Saxena deterministic test for primality. Chapter 3 introduces various algorithms, particularly the Elliptic Curve Method (ECM), the Quadratic Sieve (QS) and the Number Field Sieve (NFS) for integer factorization. This chapter also discusses some other computational problems that are related to factoring, such as the square root problem, the discrete logarithm problem and the quadratic residuosity problem. The final chapter presents the applications of the problems/techniques of primality testing, integer factorization, square roots, discrete logarithms and quadratic residuosity in public-key cryptography. Primality Testing and Integer Factorization in Public-Key Cryptography is designed for a professional audience composed of researchers and practitioners in industry. This book is also suitable as a secondary text for graduate-level students in computer science, mathematics and engineering.