Cryptography relies on assumptions of hardness of certain mathematical computations. While a would-be attacker sometimes has means to access data that should remain confidential, it is clear that attacking the these purportedly hard mathematical problems is an achievement whose reach is on a different scale. An undersized key for an asymmetric system could jeopardize the security of millions of users. In this talk, we discuss several recent (and also less recent) computations we did toassess the feasibility limit of the attacks on these mathematical problems. We are interested in particular on problems such as integer factoring and the discrete logarithm problem over finite fields. These two problem underpin most of today's public-key cryptography. We discuss the algorithms used, and the computational challenges associated to these attacks.

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