Group Action-based generalized perfect nonlinearity, contribution to the foundations of cryptographic solidity

• Main Author: Poinsot, Laurent
• Other Authors: Institut des Sciences de l'Ingénieur de Toulon et du Var (ISITV), Université de Toulon (UTLN), Université du Sud Toulon Var, Harari Sami
• Format: Doctoral or Postdoctoral Thesis
• Language: French
• Published: HAL CCSD 2005
• Subjects: cryptography; perfect nonlinear functions; bent functions; group actions; difference sets; linear representations; harmonic analysis; cryptographie; fonctions parfaitement non linéaires; fonctions courbes; actions de groupe; ensembles à différences; représentations linéaires; analyse harmonique; [MATH]Mathematics [math]; sami
• Online Access: https://tel.archives-ouvertes.fr/tel-00010216; https://tel.archives-ouvertes.fr/tel-00010216/document; https://tel.archives-ouvertes.fr/tel-00010216/file/tel-00010216.pdf

Mme Anne CANTEAUT, Chargée de Recherche INRIA, INRIA Rocquencourt, Présidente du jury. M. Jean-Michel COMBES, Professeur des Universités, Université du Sud Toulon-Var, Examinateur. M. James A. DAVIS, Richardson Professor of Mathematics, University of Richmond, Rapporteur. M. Sami HARARI, Professeur des Universités, Université du Sud Toulon-Var, Directeur de thèse. M. Jean-François MAURRAS, Professeur des Universités, Université de la Méditerranée, Rapporteur. M. François RODIER, Directeur de Recherche CNRS, CNRS - Institut de Mathématiques de Luminy. Notions of perfect nonlinearity and bent functions are particularly relevant in cryptography because they formalize maximal resistances against the very efficient differential and linear attacks. This thesis is then dedicated to the study of these cryptographic objects. We naturally interpret these notions in a more abstract and theoretical framework essentially by the substitution of the translations which occur in the definition of perfect nonlinearity by any group action. The properties of these actions as fidelity and regularity allow to decline this new concept into several alternatives. We develop as well its dual characterization using the Fourier transform that leads to an adapted notion of bentness. In particular in the case of a non Abelian group action, we use the linear representations theory to establish a dual matrix version. Furthermore, following the same principle, we generalize those combinatorics objects called difference sets which characterize perfect nonlinearity of functions with values in the finite field with two elements. This allows us to exhibit some constructions of functions which satisfy our generalized criteria, in particular in those cases where bent functions in the usual sense do not exist. Les notions de fonctions parfaitement non linéaires et courbes sont particulièrement pertinentes en cryptographie puisqu'elles formalisent les résistances maximales face aux très efficaces attaques différentielle et linéaire. Cette thèse est ...