An efficient characterization of a family of hyper-bent functions with multiple trace terms
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, h...
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doaj-11751e0a2a104bfead75943fbff2c1e42021-09-06T19:40:43ZengDe GruyterJournal of Mathematical Cryptology1862-29761862-29842013-07-0171436810.1515/jmc-2012-0013An efficient characterization of a family of hyper-bent functions with multiple trace termsFlori Jean-Pierre0Mesnager Sihem1Institut Télécom, Télécom ParisTech, UMR 7539, CNRS LTCI, 46 rue Barrault, 75634 Paris Cedex 13, FranceLaboratoire Analyse, Géometrie et Applications, UMR 7539, CNRS, Department of Mathematics, University of Paris XIII and University of Paris VIII, 2 rue de la liberté, 93526 Saint-Denis Cedex, FranceThe 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.https://doi.org/10.1515/jmc-2012-0013boolean functionswalsh–hadamard transformmaximum nonlinearityhyper-bent functionshyperelliptic curvesdickson polynomials |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Flori Jean-Pierre Mesnager Sihem |
spellingShingle |
Flori Jean-Pierre Mesnager Sihem An efficient characterization of a family of hyper-bent functions with multiple trace terms Journal of Mathematical Cryptology boolean functions walsh–hadamard transform maximum nonlinearity hyper-bent functions hyperelliptic curves dickson polynomials |
author_facet |
Flori Jean-Pierre Mesnager Sihem |
author_sort |
Flori Jean-Pierre |
title |
An efficient characterization of a family of hyper-bent functions with multiple trace terms |
title_short |
An efficient characterization of a family of hyper-bent functions with multiple trace terms |
title_full |
An efficient characterization of a family of hyper-bent functions with multiple trace terms |
title_fullStr |
An efficient characterization of a family of hyper-bent functions with multiple trace terms |
title_full_unstemmed |
An efficient characterization of a family of hyper-bent functions with multiple trace terms |
title_sort |
efficient characterization of a family of hyper-bent functions with multiple trace terms |
publisher |
De Gruyter |
series |
Journal of Mathematical Cryptology |
issn |
1862-2976 1862-2984 |
publishDate |
2013-07-01 |
description |
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. |
topic |
boolean functions walsh–hadamard transform maximum nonlinearity hyper-bent functions hyperelliptic curves dickson polynomials |
url |
https://doi.org/10.1515/jmc-2012-0013 |
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