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bent functions on Finite Nonabelian Groups and Relative Difference Sets

JOURNAL OF COMBINATORIAL DESIGNS 2025年第4期33卷 125-136页

作者： Xu, Bangteng Eastern Kentucky Univ Dept Math & Stat Richmond KY 40475 USA

It is well known that the perfect nonlinearity of a function between finite groups G $G$ and H $H$ can be characterized by its graph in terms of relative difference set in the direct product GxH $G\times H$ (cf. [4]). Let T $T$ be the infinite set of complex roots of unity. A T $T$-valued function f $f$ on an arbitrary finite group G $G$ is associated with a finite cyclic subgroup Tf ${T}_{f}$ in the multiplicative group of nonzero complex numbers. For a bent function f $f$ on G $G$ in general, its graph is not a relative difference set in the direct product GxTf $G\times {T}_{f}$. In this paper, we investigate the necessary and sufficient conditions under which the graph of a bent function f $f$ on G $G$ is a relative difference set in GxTf $G\times {T}_{f}$. Cyclotomic fields and their integral bases play an important role in our discussions.

关键词： bent functions cyclotomic fields Fourier transforms perfect nonlinear functions relative difference sets

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A new method of constructing (k plus s)-variable bent functions based on a family of s-plateaued functions on k variables

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DESIGNS CODES AND CRYPTOGRAPHY 2025年第3期93卷 443-465页

作者： Su, Sihong Chen, Xiaoyan Henan Univ Sch Math & Stat Kaifeng 475004 Peoples R China Henan Univ Ctr Appl Math Henan Prov Zhengzhou 450046 Peoples R China

It is important to study the new construction methods of bent functions. In this paper, we first propose a secondary construction method of (k + s)-variable bent function g through a family of s-plateaued functions f0, f1,..., f2s- 1 on k variables with disjointWalsh supports, which can be obtained through any given (k - s)-variable bent function f by selecting 2s disjoint affine subspaces S0, S1,..., S2s- 1 of Fk 2 with dimension k - s to specify the Walsh support of these s-plateaued functions respectively, where s is a positive integer and k - s is a positive even integer. The dual functions of these newly constructed bent functions are determined. This secondary construction method of bent functions has a great improvement in counting. As a generalization, we find that the one initial (k - s)-variable bent function f can be replaced by several different (k - s)-variable bent functions. Compared to the first construction method, the latter one gives much more bent functions. It is worth mentioning that it can give all the 896 bent functions on 4 variables.

关键词： Walsh-Hadamard transform bent functions Plateaued functions Dual functions Disjoint spectra functions

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bent functions and line ovals

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FINITE FIELDS AND THEIR APPLICATIONS 2017年 47卷 94-124页

作者： Abdukhalikov, Kanat UAE Univ Dept Math Sci POB 15551 Al Ain U Arab Emirates

In this paper we study those bent functions which are linear on elements of spreads, their connections with ovals and line ovals, and we give descriptions of their dual bent functions. In particular, we give a geometric characterization of Niho bent functions and of their duals, we give explicit formula for the dual bent function and present direct connections with ovals and line ovals. We also show that bent functions which are linear on elements of inequivalent spreads can be EA equivalent. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Spreads Ovals Line ovals Quasifields Semifields bent functions Niho bent functions

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bent functions From Involutions Over F_2n

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第4期64卷 2979-2986页

作者： Coulter, Robert S. Mesnager, Sihem Univ Delaware Dept Math Sci Newark DE 19716 USA Univ Paris VIII Dept Math F-93526 St Denis France Univ Paris XIII LAGA CNRS Sorbonne Paris CiteUMR 7539 F-93430 Villetaneuse France Telecom ParisTech F-75013 Paris France

bent functions are maximally nonlinear Boolean functions. Introduced by Rothaus and first examined by Dillon, these important functions have subsequently been studied by many researchers over the last four decades. Since a complete classification of bent functions appears elusive, many researchers concentrate on methods for constructing bent functions. In this paper, we investigate constructions of bent functions from involutions over finite fields in even characteristic. We present a generic construction technique, study its equivalence issues and show that linear involutions (which are an important class of permutations) over finite fields give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions, whose dual functions are directly obtained without computation. The existence of bent functions from involutions relies heavily on solving systems of equations over finite fields.

关键词： Galois fields permutations involutions bent functions symmetric cryptography coding theory

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bent functions with 2′ Niho exponents

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IEEE TRANSACTIONS ON INFORMATION THEORY 2006年第12期52卷 5529-5532页

作者： Leander, Gregor Kholosha, Alexander Ruhr Univ Bochum D-44780 Bochum Germany Univ Bergen Dept Informat Selmer Ctr N-5020 Bergen Norway

In this note, a new primary construction of bent functions consisting of a linear combination of 2(r) Niho exponents is presented. This construction generalizes one of the constructions proven by H. Dobbertin, G. Lean... 详细信息

关键词： bent functions monomial Boolean functions trace expansion

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bent functions of Maximal Degree

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IEEE TRANSACTIONS ON INFORMATION THEORY 2012年第2期58卷 1186-1190页

作者： Cesmelioglu, Ayca Meidl, Wilfried Sabanci Univ MDBF TR-34956 Istanbul Turkey

In this paper, a technique for constructing p-ary bent functions from plateaued functions is presented. This generalizes earlier techniques of constructing bent from near-bent functions. The Fourier spectrum of quadratic monomials is analyzed, and examples of quadratic functions with highest possible absolute values in their Fourier spectrum are given. Applying the construction of bent functions to the latter class of functions yields bent functions attaining upper bounds for the algebraic degree when p = 3,5. Until now, no construction of bent functions attaining these bounds was known.

关键词： Algebraic degree bent functions Fourier transform plateaued functions quadratic functions

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bent functions from nonlinear permutations and conversely

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN functions AND SEQUENCES 2019年第2期11卷 207-225页

作者： Pasalic, Enes Hodzic, Samir Zhang, Fengrong Wei, Yongzhuang Univ Primorska FAMNIT Koper 6000 Slovenia Univ Primorska IAM Koper 6000 Slovenia China Univ Min & Technol Sch Comp Sci & Technol Xuzhou 221116 Jiangsu Peoples R China Guangxi Key Lab Cryptog & Informat Secur Guilin 5413004 Peoples R China Guilin Univ Elect Technol Guangxi Key Lab Wireless Wideband Commun & Signal Guilin 541004 Peoples R China

This work extends the idea introduced by Hou and Langevin (J. Combin. Theory, Ser. A, 80:232-246, 1997) of applying nonlinear permutations to (a portion of) the input variable space of a given Boolean function so that the resulting function is bent. Applying such a permutation to a bent function that can be represented in a suitable form then gives an affine inequivalent bent function which potentially does not belong to the same class as the original one. While Hou and Langevin only provided two sporadic examples of bent functions that can be turned into affine inequivalent ones, in this article we identify two generic families of bent functions suitable for generating such affine inequivalent counterparts. The same method when applied to the Marioana-McFarland class of bent functions, depending on the subset of inputs to which a nonlinear action is applied, either lead to bent functions that are provably within the same class or to bent functions that are potentially outside this class. The problem of finding suitable permutations that act nonlinearly on more than two input variables of the initial function and ensure the bentness of the resulting function appears to be generally hard. In this direction, we only slightly extend the approach of Hou and Langevin by identifying suitable permutations that act nonlinearly on three input variabl es. Most notably, the existence of nonlinear permutations that act without strict separation of the input space in terms of linear and nonlinear action is also confirmed. Finally, we show a direct correspondence between (some classes of) bent functions and permutations by providing an efficient method to define permutations using the derivatives of a given bent function. This not only gives a relationship between two seemingly different algebraic objects, but also provides us with a new infinite family of permutations over finite fields.

关键词： Nonlinear permutations bent functions Maiorana-McFarland class Affine equivalence

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bent functions linear on elements of some classical spreads and presemifields spreads

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN functions AND SEQUENCES 2017年第1期9卷 3-21页

作者： Abdukhalikov, Kanat Mesnager, Sihem UAE Univ Dept Math Sci POB 15551 Al Ain U Arab Emirates Univ Paris XIII Univ Paris VIII CNRS LAGADept MathLAGAUMR 7539 Paris France Telecom ParisTech Paris France

bent functions are maximally nonlinear Boolean functions with an even number of variables. They have attracted a lot of research for four decades because of their own sake as interesting combinatorial objects, and also because of their relations to coding theory, sequences and their applications in cryptography and other domains such as design theory. In this paper we investigate explicit constructions of bent functions which are linear on elements of spreads. After presenting an overview on this topic, we study bent functions which are linear on elements of presemifield spreads and give explicit descriptions of such functions for known commutative presemifields. A direct connection between bent functions which are linear on elements of the Desarguesian spread and oval polynomials over finite fields was proved by Carlet and the second author. Very recently, further nice extensions have been made by Carlet in another context. We introduce oval polynomials for semifields which are dual to symplectic semifields. In particular, it is shown that from a linear oval polynomial for a semifield one can get an oval polynomial for transposed semifield.

关键词： Boolean functions bent functions Walsh Hadamard transform Spreads Quasifields Semifields Symplectic semifields Oval polynomials

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bent functions in the partial spread class generated by linear recurring sequences

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DESIGNS CODES AND CRYPTOGRAPHY 2023年第1期91卷 63-82页

作者： Gadouleau, Maximilien Mariot, Luca Picek, Stjepan Univ Durham Dept Comp Sci South Rd Durham DH1 3LE England Radboud Univ Nijmegen Digital Secur Grp POB 9010 NL-6500 GL Nijmegen Netherlands

We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials are either 1 or 2. We then count the resulting sets of polynomials and prove that, for degree 1, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all PS- and PS+ bent functions of n = 8 variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree d = 2 are not EA-equivalent to any Maiorana-McFarland or Desarguesian partial spread function.

关键词： bent functions Partial spreads Cyclic codes Linear recurring sequences Polynomials

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Cubic bent functions

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DISCRETE MATHEMATICS 1998年第1-3期189卷 149-161页

作者： Hou, XD Wright State Univ Dept Math & Stat Dayton OH 45435 USA

For any Boolean function f on GF(2)(m), we define a sequence of ranks r(i)(f), 1 less than or equal to i less than or equal to m, which are invariant under the action of the general linear group GL(m, 2). If f is a cubic bent function in 2k variables, we show that when r(3)(f) less than or equal to k, f is either obtained from a cubic bent function in 2k - 2 variables, or is in a well-known family of bent functions. We also determine all cubic bent functions in eight variables. (C) 1998 Elsevier Science B.V. All rights reserved.

关键词： bent functions Reed-Muller codes

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