×

Automorphisms and isomorphisms of some \(p\)-ary bent functions. (English) Zbl 1456.94144

In this continuation of the author’s paper [Commun. Algebra 34, No. 3, 1077–1131 (2006; Zbl 1085.05019)] on Boolean (bent) functions, \(p\)-ary bent functions are similarly investigated. EA-equivalence of (bent) functions is in general not easy to decide. Simple invariants, like algebraic degree, are usually not sufficient to decide equivalence of bent functions, stronger methods seem necessary.
In this paper, group-theoretic methods are applied to analyse equivalence of (some classes of) \(p\)-ary bent functions. For a given (bent) function \(f\) from an \(n\)-dimensional vector space \(V\) over \(\mathbb{F}_p\) to \(\mathbb{F}_p\), the author considers the group \(\mathbf{EA}(f)\) of EA automorphisms, i.e. \(\phi_{11} \in (V)\), \(\phi_{22} \in \mathrm{GL}(\mathbb{F}_p)\), \(\phi_{12} \in \operatorname{Hom}(V,\mathbb{F}_p)\) and \(v\in V, w\in \mathbb{F}_p\) such that
\[ f(\phi_{11}(x)+v) = \phi_{22}(f(x)) + \phi_{12}(x) + w \quad\text{for all }x\in V. \]
The structure of this group is invariant under EA-equivalence.
As another invariant under EA-equivalence for \(p\)-ary functions, the author suggests the set \(\{v\in V, D_v^2f = 0\}\), where \(D_vf(x) = f(x+v)-f(x)\) denotes the derivative of \(f\) in direction \(v\).
In the first part, the author discusses a secondary construction of three types of (non-quadratic) bent functions \(f\), describes \(\mathbf{EA}(f)\) and as a consequence solves the equivalence problem for these types of bent functions.
In the second part of the paper, \(\mathbf{EA}(f)\) is described for Maiorana-McFarland bent functions of the form \(\mathrm{Tr}_n(xy^l)\), \(\gcd(l,p^n-1) = 1\), and the question when two such bent functions are EA-equivalent is solved.

MSC:

94D10 Boolean functions
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
06E30 Boolean functions
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

Citations:

Zbl 1085.05019

Software:

GAP; Magma
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aschbacher, M., Finite Group Theory (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 0965.20009
[2] Bell, G., On the cohomology of the finite special linear groups, I, J. Algebra, 54, 216-238 (1978) · Zbl 0389.20043 · doi:10.1016/0021-8693(78)90027-3
[3] Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system I: the user language, J. Symbolic Comput., 24, 235-265 (1997) · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125
[4] Budaghyan, L.; Carlet, C., CCZ-equivalence of bent vectorial functions and related constructions, Des. Codes Cryptogr., 59, 69-87 (2011) · Zbl 1215.94035 · doi:10.1007/s10623-010-9466-9
[5] Cameron, P., Primitive permutation groups and finite simple groups, Bull. Lond. Math. Soc., 13, 1-22 (1981) · Zbl 0463.20003 · doi:10.1112/blms/13.1.1
[6] Carter, R., Simple Groups of Lie Type (1989), Hoboken: Wiley, Hoboken · Zbl 0723.20006
[7] Cesmelioglu, A.; Meidl, W.; Pott, A., Generalized Maiorana-McFarland class and the normality of \(p\)-ary bent functions, Finite Fields Appl., 24, 105-117 (2013) · Zbl 1305.11112 · doi:10.1016/j.ffa.2013.06.001
[8] Dempwolff, U., A characterization of the generalized twisted field planes, Arch. Math., 50, 477-480 (1988) · Zbl 0619.51005 · doi:10.1007/BF01196512
[9] Dempwolff, U., Automorphisms and equivalence of bent functions and of difference sets in elementary abelian 2-groups, Comm. Algebra, 34, 3, 1077-1131 (2006) · Zbl 1085.05019 · doi:10.1080/00927870500442062
[10] Dempwolff, U., Some doubly transitive bilinear dual hyperovals and their ambient spaces, European J. Combin., 44, 1-22 (2015) · Zbl 1341.51008 · doi:10.1016/j.ejc.2014.09.003
[11] Dempwolff, U., The automorphism groups of doubly transitive bilinear dual hyperovals, Adv. Geom., 17, 91-108 (2017) · Zbl 1441.20002 · doi:10.1515/advgeom-2016-0030
[12] Dempwolff, U., CCZ equivalence of power functions, Des. Codes Cryptogr., 86, 665-692 (2018) · Zbl 1426.11132 · doi:10.1007/s10623-017-0350-8
[13] Gangopadhyay, S., Affine inequivalence of cubic Maiorana-McFarland type bent functions, Discrete Appl. Math., 161, 1141-1146 (2013) · Zbl 1279.06010 · doi:10.1016/j.dam.2012.11.017
[14] Grove, L.: Classical Groups and Geometric Algebra. AMS Series Graduate Studies in Mathematics, vol. 39. Providence, RI (2002)
[15] Harris, M., A note on the classical groups over finite fields, Rev. Roumaine Math. Pures Appl., 27, 2, 159-167 (1982) · Zbl 0487.20032
[16] Huppert, Bertram, Permutationsgruppen und lineare Gruppen, Grundlehren der mathematischen Wissenschaften, 144-250 (1967), Berlin, Heidelberg: Springer Berlin Heidelberg, Berlin, Heidelberg · Zbl 0217.07201
[17] JONES, WAYNE; PARSHALL, BRIAN, ON THE 1-COHOMOLOGY OF FINITE GROUPS OF LIE TYPE, Proceedings of the Conference on Finite Groups, 313-328 (1976) · Zbl 0345.20046
[18] Kantor, W., Linear groups containing a Singer cycle, J. Algebra, 62, 232-234 (1980) · Zbl 0429.20004 · doi:10.1016/0021-8693(80)90214-8
[19] Kumar, P.; Scholz, R.; Welch, L., Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40, 90-107 (1985) · Zbl 0585.94016 · doi:10.1016/0097-3165(85)90049-4
[20] Liebeck, M., The affine permutation groups of rank three, Proc. Lond. Math. Soc. (3), 54, 477-516 (1987) · Zbl 0621.20001 · doi:10.1112/plms/s3-54.3.477
[21] Lübeck, F., Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math., 4, 135-169 (2001) · Zbl 1053.20008 · doi:10.1112/S1461157000000838
[22] Schneider, L., Minimale Darstellungen endlicher klassischer Gruppen in natürlicher Charakteristik (2004), Göttingen: Cuvillier Verlag, Göttingen
[23] Sin, P.: Personal communication
[24] Steinberg, R.: Lectures on Chevalley Groups. AMS-University Lecture Series, vol. 66. Providence, RI (2016) · Zbl 1361.20003
[25] The GAP Group: GAP-Groups Algorithms and Programming, version 4.4.12 (2008). http://www.gap-system.org. Accessed 15 Jan 2018
[26] Völklein, F., The 1-Cohomology of the adjoint module of a Chevalley group, Forum Math., 1, 1-13 (1989) · Zbl 0649.20042 · doi:10.1515/form.1989.1.1
[27] Zsigmondy, K., Zur Theorie der Potenzreste, Monatshefte Math. Phys., 3, 3, 265-284 (1892) · JFM 24.0176.02 · doi:10.1007/BF01692444
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.