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The group of automorphisms of the set of self-dual bent functions. (English) Zbl 07371093
Summary: A bent function is a Boolean function in even number of variables which is on the maximal Hamming distance from the set of affine Boolean functions. It is called self-dual if it coincides with its dual. It is called anti-self-dual if it is equal to the negation of its dual. A mapping of the set of all Boolean functions in \(n\) variables to itself is said to be isometric if it preserves the Hamming distance. In this paper we study isometric mappings which preserve self-duality and anti-self-duality of a Boolean bent function. The complete characterization of these mappings is obtained for \(n\geqslant 4\). Based on this result, the set of isometric mappings which preserve the Rayleigh quotient of the Sylvester Hadamard matrix, is characterized. The Rayleigh quotient measures the Hamming distance between bent function and its dual, so as a corollary, all isometric mappings which preserve bentness and the Hamming distance between bent function and its dual are described.
MSC:
94D10 Boolean functions
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[1] Carlet, C.: Boolean functions for cryptography and error correcting code. In: Crama, Y., Hammer, P.L. (eds.) Boolean models and methods in mathematics, computer science, and engineering, pp 257-397. Cambridge University Press, Cambridge (2010) · Zbl 1209.94035
[2] Carlet, C.; Danielson, LE; Parker, MG; Solé, P., Self-dual bent functions, Int. J. Inform. Coding Theory, 1, 384-399 (2010) · Zbl 1204.94118
[3] Carlet, C.; Mesnager, S., Four decades of research on bent functions, Journal Des. Codes Cryptogr. Springer, 78, 1, 5-50 (2016) · Zbl 1378.94028
[4] Cusick, TW; Stănică P., Cryptographic Boolean functions and applications, 288-288 (2017), London: Acad. Press, London
[5] Danielsen, LE; Parker, MG; Solé, P., The Rayleigh quotient of bent functions, Springer Lect. Notes in Comp. Sci. 5921, 418-432 (2009), Berlin: Springer, Berlin · Zbl 1234.06010
[6] Dillon, J., Elementary Hadamard Difference Sets, PhD. dissertation (1974), Univ Maryland: College Park, Univ Maryland · Zbl 0346.05003
[7] Feulner, T.; Sok, L.; Solé, P.; Wassermann, A., Towards the classification of self-dual bent functions in eight variables, Des. Codes Cryptogr., 68, 1, 395-406 (2013) · Zbl 1280.94053
[8] Hou, X-D, Classification of self dual quadratic bent functions, Des Codes Cryptogr., 63, 2, 183-198 (2012) · Zbl 1264.06021
[9] Hyun, JY; Lee, H.; Lee, Y., MacWilliams duality and Gleason-type theorem on self-dual bent functions, Des Codes Cryptogr., 63, 3, 295-304 (2012) · Zbl 1259.94071
[10] Janusz, GJ, Parametrization of self-dual codes by orthogonal matrices, Finite Fields Appl., 13, 3, 450-491 (2007) · Zbl 1138.94389
[11] Kutsenko, AV, The Hamming distance spectrum between self-dual Maiorana-McFarland bent functions, J. Appl. Ind. Math., 12, 1, 112-125 (2018) · Zbl 1413.94045
[12] Kutsenko, A., Metrical properties of self-dual bent functions, Des. Codes Cryptogr., 88, 1, 201-222 (2020) · Zbl 07149379
[13] Luo, G.; Cao, X.; Mesnager, S., Several new classes of self-dual bent functions derived from involutions, Cryptogr. Commun., 11, 6, 1261-1273 (2019) · Zbl 1460.11147
[14] MacWilliams, FJ; Sloane, NJA, The theory of error correcting codes (1977), Amsterdam: North-Holland, Amsterdam
[15] Markov A.A.: On transformations without error propagation. In: Selected works, Vol. II: Theory of algorithms and constructive mathematics. Mathematical Logic. Informatics and Related Topics, p. 70-93, MTsNMO, Moscow [Russian] (2003)
[16] Mesnager, S., Several new infinite families of bent functions and their duals, IEEE Trans. Inf. Theory, 60, 7, 4397-4407 (2014) · Zbl 1360.94480
[17] Mesnager, S., Bent functions: Fundamentals and results, 544-544 (2016), Berlin: Springer, Berlin · Zbl 1364.94008
[18] Rothaus, OS, On bent functions, J. Combin. Theory. Ser. A, 20, 3, 300-305 (1976) · Zbl 0336.12012
[19] Sok, L.; Shi, M.; Solé, P., Classification and Construction of quaternary self-dual bent functions, Cryptogr. Commun., 10, 2, 277-289 (2018) · Zbl 1412.94257
[20] Tokareva, N., The group of automorphisms of the set of bent functions, Discret. Math. Appl., 20, 5, 655-664 (2010) · Zbl 1211.94057
[21] Tokareva, N.: Bent functions: Results and applications to cryptography, 230 p., Acad. Press Elsevier (2015) · Zbl 1372.94002
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