Integer factorization and discrete logarithm problems

These are notes for a lecture given at CIRM in 2014, for the Journées Nationales du Calcul Formel . We explain the basic algorithms based on combining congruences for solving the integer factorization and the discrete logarithm problems. We highlight two particular situations where the interaction with symbolic computation is visible: the use of Gröbner basis in Joux's algorithm for discrete logarithm in nite eld of small characteristic, and the exact sparse linear algebra tools that occur in the Number Field Sieve algorithm for discrete logarithm in large characteristic. Disclaimer: These notes contain no new material. They also do not aim to be a survey, and exhaustivity is not a goal. As a consequence, many important works (old or recent) are not mentioned. 1 The problems and their cryptographic signi cance 1.1 Cryptographic context Cryptography, as a science that studies tools to secure communications in a wide sense, contains various sub-themes among which the so-called public-key cryptography covers algorithms such as digital signatures, and asymmetric encryption, i.e. encryption systems in which the key that is used to encrypt is publicly known, while the corresponding decryption key is known only by the recipient. This is useful in particular when the participants do not share a common secret prior to any interaction. Most public key algorithms used today are based on di cult problems in number theory, and more precisely integer factorization and discrete logarithm problems, which constitute the central topic of this lecture. Let us give two examples. EMV. The EMV is the standard adopted by Visa, Mastercard, the French Carte Bleue and many others, to secure the chip-and-pin smartcard transactions. The public key part of the standard is based on the RSA cryptosystem, whose security relies on the presumed di culty of factoring integers. The typical key sizes found in today's cards are 1024 bits or slightly more. Anyone who would be able to factor numbers of this size could easily break into the system and create fake transactions. SSL/TLS. The SSL/TLS standard is the one used to secure many protocols found on the internet. For instance, the s at the end of https acronym means that the connection between your webbrowser and the server is secured using SSL/TLS. By clicking on the lock picture when browsing