An efficient characterization of a family of hyper-bent functions with multiple trace terms

• Main Authors: Flori Jean-Pierre, Mesnager Sihem
• Format: Article
• Language: English
• Published: De Gruyter 2013-07-01
• Series: Journal of Mathematical Cryptology
• Subjects: boolean functions; walsh–hadamard transform; maximum nonlinearity; hyper-bent functions; hyperelliptic curves; dickson polynomials
• Online Access: https://doi.org/10.1515/jmc-2012-0013

The connection between exponential sums and algebraic varieties has been known for at least six decades. Recently, Lisoněk exploited it to reformulate the Charpin–Gong characterization of a large class of hyper-bent functions in terms of numbers of points on hyperelliptic curves. As a consequence, he obtained a polynomial time and space algorithm for certain subclasses of functions in the Charpin–Gong family. In this paper, we settle a more general framework, together with detailed proofs, for such an approach and show that it applies naturally to a distinct family of functions proposed by Mesnager. Doing so, a polynomial time and space test for the hyper-bentness of functions in this family is obtained as well. Nonetheless, a straightforward application of such results does not provide a satisfactory criterion for explicit generation of functions in the Mesnager family. To address this issue, we show how to obtain a more efficient test leading to a substantial practical gain. We finally elaborate on an open problem about hyperelliptic curves related to a family of Boolean functions studied by Charpin and Gong.