Development of number factorization tools and impacts on asymmetric cryptography

The aim of this thesis is to introduce readers to factorization of large numbers, describe the issue of factorization and its methods and analyze their impact on asymmetric cryptography, especially on RSA. The thesis first explains the RSA encryption algorithm, principles of its operation and security. Then it deals with the problem of factorization. It describes both methods that were used long times ago, namely the trial division algorithm, Fermat method and Euler algorithm, as well as their modern successors, namely Pollard p-1 method, Lenstra method, Dixon factorization method, quadratic sieve and general number field sieve. It also describes how quantum computers work, their purpose and use in Shor's factorization algorithm. In the last part of the thesis, the impact of factorization on RSA and RSA key length is discussed. It analyzes keys that have already been broken, those that should be used now. Finally, it describes the threat of quantum computers to break the RSA almost independent of key length and explains how real it is today.