MATHEMATICAL BACKGROUND OF PUBLIC KEY CRYPTOGRAPHY
Gerhard Frey, Tanja Lange · TU/e Research Portal · 2005
Abstract. — The two main systems used for public key cryptography are RSA and protocols based on the discrete logarithm problem in some cyclic group. We focus on the latter problem and state cryptographic protocols and mathematical background material. Résumé (Éléments mathématiques de la cryptographie à clef publique). — Les deux systèmes principaux de cryptographie à clef publique sont RSA et le calcul de logarithmes discrets dans un groupe cyclique. Nous nous intéressons aux logarithmes discrets et présentons les faits mathématiques qu’il faut connaître pour apprendre la cryptographie mathématique. 1. Data Security and Arithmetic Cryptography is, in the true sense of the word, a classic discipline: we find it in Mesopotamia and Caesar used it. Typically, the historical examples involve secret services and military. Information is exchanged amongst a limited community in which each member is to be trusted. Like Caesar’s chiffre these systems were entirely symmetric. Thus, the communicating parties needed to have a common key which is used to de- and encrypt. The key exchange posed a problem (and gives a marvellous plot for spy-novels) but the number of people involved was rather bounded. This has changed dramatically because of electronic communication in public networks. Since 2000 Mathematics Subject Classification. — 11T71. Key words and phrases. — Elliptic curve cryptography, mathematics of public key cryptography, hyperelliptic curves. The authors would like to thank the organizers of the conference for generous support, an interesting program and last but not least for a very inspiring and pleasant atmosphere. The second author acknowledges financial support by STORK