The recent succes of collision attacks on cryptographic hash functions have created some turmoil in the cryptologic research community. In particular, collisions for MD5 and SHA-0 have been exhibited, and some costly but feasible attacks on SHA-1 have been described. This represent a potential threat for some cryptographic applications, as digital signatures. We propose two main topics for our project. 1) Understand the best attacks, in particular those presented by the Chinese researcher Wang. Actually, she provides some equations that allow the construction of collisions. It is possible to check the validity of those equations, but the way they are obtained is not very clear and deserves to be carefully examined and may hopefully help to define design criteria. 2) Find new design techniques for secure hash functions. In particular, we propose to extend some recent works on hash functions with security reduction. By combining those functions using constructions `a la Merkle-Damgard, we hope to produce fast cryptographic hash functions with a proof of security.