Modular computations involved in public key cryptography applications most often use a standardized prime modulo, the choice of which is not always free in practice. The improvement of modular operations is fundamental for the efficiency and safety of these primitives. This thesis proposes to provide an efficient modular arithmetic for the largest possible number of primes, while protecting it against certain types of attacks. For this purpose, we are interested in the PMNS system used for modular arithmetic, and propose methods to obtain many PMNS for a given prime, with an efficient arithmetic on the representations. We also consider the randomization of modular computations via algorithms of type Montgomery and Babaï by exploiting the intrinsic redundancy of PMNS. Induced changes of data representation during the calculation prevent an attacker from making useful assumptions about these representations. We then present a hybrid system, HyPoRes, with an algorithm that improves modular reductions for any prime modulo. The numbers are represented in a PMNS with coefficients in RNS. The modular reduction is faster than in conventional RNS for the primes standardized for ECC. In parallel, we are interested in a type of representation used to compute real solutions of fuzzy systems. We revisit the global approach of resolution using classical algebraic techniques and strengthen it. These results include a real system called the real transform that simplifies computations, and the management of the signs of the solutions.
Defence : 12/06/2019 - 10h - Campus Jussieu, salle Jacques Pitrat (25-26/105) Jury members : Mme Marine Minier, Professeure, Université de Lorraine [rapporteur]
M. Clément Pernet, Maître de conférences HDR, Université Grenoble Alpes [rapporteur]
M. Lokmane Abbas-Turki, Maître de conférences, Sorbonne Université
M. Jean-Claude Bajard, Professeur, Sorbonne Université
M. Louis Goubin (examinateur), Professeur, UVSQ
Mme Annick Valibouze, Professeure, Sorbonne Université
Ph. Aubry, J. Marrez, A. Valibouze : “The Real Transform: Computing Positive Solutions of Fuzzy Polynomial Systems”, 11th International Conference on Fuzzy Computation Theory and Applications, vol. 1: FCTA, Proceedings of the 11th International Joint Conference on Computational Intelligence, Vienna, Austria, pp. 351-359, (SciTePress - Science and Technology Publications) (2019)