**Objective**: This course is of the “selected passages” kind in view of the scope of the field covered by so-called discrete mathematics. This short 12-session version will be an introduction to cryptography and to unstable dynamic systems. This introduction should enable students to be more conscious, on the one hand, of the problems and difficulties related to computer security, and on the other hand, to the complex systems, of which the non-linear ones make up a sizeable category.

**Programme**: The first part is an introduction to modern cryptography and its links with the theory of numbers and complexity. After a working description of the RSA system, the most commonly used public key system, we will present its principle features: difficulty of factorization, primality, single-direction function. Studying and understanding these various points require knowledge of some of the results and notions in the number theory (density and characterization of prime numbers, modular exponential, etc.) and in complexity (classes P, NP, RP, etc.).

The second part will focus on another field in which the discrete aspect plays a determining role: dynamic systems. We shall see how, on the basis of a continuous description (differential equations) the asymptotic behaviour of solutions can be understood naturally with discrete tools. In addition to the well-known case of resonance phenomena (theory of non-linear oscillations), we shall also refer to phenomena linked to the regularity of functions, themselves linked to arithmetical problems as in the case of certain periodic or delayed-change solutions. We shall also provide a detailed description of a preliminary simple and totally discrete model whereby the main features of chaotic dynamics can be understood.

In the presentation, we shall highlight the practical and fundamental motivations of the notions and concepts presented. Use will be made every time of specific examples. Demonstrations will be resorted to purely for their pedagogical value. Abstract developments will be kept to a strict minimum.

**Last Modification :**Saturday 11 December 2010