The group of automorphisms of the set of self-dual bent functions
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The group of automorphisms of the set of self-dual bent functions Aleksandr Kutsenko1,2 Received: 25 August 2019 / Accepted: 6 May 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 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. Keywords Boolean functions · Self-dual bent · Isometric mappings · The group of automorphisms · The Rayleigh quotient
1 Introduction The term “bent function” was introduced by Oscar Rothaus in the 1960s [18]. It is known [21], that at the same time Boolean functions with maximal nonlinearity were also studied in the Soviet Union. The term minimal function, which is actually a counterpart of a bent function, was proposed by the Soviet scientists Eliseev and Stepchenkov in 1962.
This article belongs to the Topical Collection: Boolean Functions and Their Applications IV Guest Editors: Lilya Budaghyan and Tor Helleseth The author was supported by the Russian Foundation for Basic Research (projects no. 18-07-01394, 2031-70043), the study was supported within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0017) and Laboratory of Cryptography JetBrains Research. Aleksandr Kutsenko
[email protected] 1
Sobolev Institute of Mathematics, Novosibirsk, Russia
2
Novosibirsk State University, Novosibirsk, Russia
Cryptography and Communications
Bent functions have connections with such combinatorial objects as Hadamard matrices and difference sets. Since bent functions have maximum Hamming distance to linear structures and affine functions they deserve attention for practical applications in symmetric cryptography, in particular, for block and stream ciphers. We refer to the survey [3] and monographies of Mesnager [17] and Tokareva [21] for more information concerning known results and open problems related to bent functions. For each bent function, its dual bent function is uniquely defined. More information about properties of dual bent functions one can find in work [3]. A bent function that coincides with its dual is called self-dual. There are a number of papers devoted to open problems including
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