A Residue Approach of the Finite Fields Arithmetics

Finite fields arithmetic is one of the challenges in current computer arithmetic. It occurs, in particular, in cryptography where the needs increase with the evolution of the technologies and also of the attacks. Through our research, we have proposed different systems based on residues representations. Different kinds of finite fields are concerned with. For each of them, some specificities of the representations are exploited to ensure the efficiency, as well as for the performances, than for the robustness to side channel attacks. In this paper, we deal with three similar approaches: the first one is dedicated to prime field using residue number systems, a second one concerns extension finite fields of characteristic two, the last one discusses of medium characteristic finite fields. The main interest of these systems is their inherent modularity, well suited for circuit implementations.