Nonlinearity of Some Invariant Boolean Functions

Abstract

One of the hardest problems in coding theory is to evaluate the covering radius of first order Reed-Muller codes RM(1, m), and more recently the balanced covering radius for cryptographical purposes. The aim of this paper is to present some new results on this subject. We mainly study boolean functions invariant under the action of some finite groups, following the idea of Patterson and Wiedemann [The covering radius of the (1, 15) Reed-Muller Code is atleast 16276, IEEE Trans Inform Theory, Vol. 29 (1983) 354.]. Our method is Fourier transforms and our results are both theoretical and numerical.

Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros

“…A more general study of Boolean functions invariant under the action of some finite groups has been presented in [17] that demonstrated 15-variable functions with nonlinearity greater than the bent concatenation bound (but not exceeding the nonlinearity reported in [25]). Our study can be considered as looking into a particular case where the functions are invariant under the action of the group of Frobenius automorphisms.…”

Non linéarité des fonctions booléennes données par des traces de polynômes de degré binaire 3

“…Pour m impair, il serait particulièrement intéressant de trouver des fonctions avec une non-linéarité plus grande que celle de fonctions booléennes quadratiques (appelées presque optimales dans [2]). Ceci a été fait dans le travail de Patterson et de Wiedemann [16] et également de Langevin et Zanotti [11] et plus récemment par Kavut, Maitra et Yücel [12]. Soit q = 2 m et k = F 2 m assimilé comme espace vectoriel sur F 2 à F m 2 .…”