Integer Factorization – Cryptology Meets Number Theory

Abstract

Integer factorization is one of the oldest mathematical problems. Initially, the interest in factorization was motivated by curiosity about be­haviour of prime numbers, which are the basic building blocks of all other integers. Early factorization algorithms were not very efficient. However, this dramatically has changed after the invention of the well-known RSA public-key cryptosystem. The reason for this was simple. Finding an efficient fac­toring algorithm is equivalent to breaking RSA. The work overviews development of integer factoring algorithms. It starts from the classical sieve of Eratosthenes, covers the Fermat algorithm and explains the quadratic sieve, which is a good representative of modern fac­toring algorithms. The progress in factoring is illustrated by examples of RSA challenge moduli, which have been factorized by groups of mathemati­cians and cryptographers. Shor's quantum factorization algorithm with poly­nomial complexity is described and the impact on public-key encryption is discussed.

Keywords

• cryptography
• number theory
• public-key cryptography
• factorization
• RSA Cryptosystems
• quantum computing