Dickson polynomials, hyperelliptic curves and hyper-bent functions

Authors:
Jean-Pierre Flori;Sihem Mesnager
Affiliations:
ANSSI (Agence nationale de la sécurité des sytèmes d'information), Paris, SP, France;LAGA (Laboratoire Analyse, Géometrie et Applications), UMR 7539, CNRS, Department of Mathematics, University of Paris XIII and University of Paris VIII, Saint-Denis Cedex, France
Venue:
SETA'12 Proceedings of the 7th international conference on Sequences and Their Applications
Year:
2012

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Abstract

In this paper, we study the action of Dickson polynomials on subsets of finite fields of even characteristic related to the trace of the inverse of an element and provide an alternate proof of a not so well-known result. Such properties are then applied to the study of a family of Boolean functions and a characterization of their hyper-bentness in terms of exponential sums recently proposed by Wang et al.Finally, we extend previous works of Lisoněk and Flori and Mesnager to reformulate this characterization in terms of the number of points on hyperelliptic curves and present some numerical results leading to an interesting problem.