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Linear codes and incidence structures of bent functions and their generalizations [article]

Wilfried Meidl and Alexandr A. Polujan and Alexander Pott
2020 arXiv   pre-print
problem for (n,m)-bent functions.  ...  In particular, we give a new design-theoretic characterization of (n,m)-plateaued and (n,m)-bent functions and provide a coding-theoretic as well as a design-theoretic interpretation of the extendability  ...  Acknowledgement Wilfried Meidl is supported by the FWF Project P 30966.  ... 
arXiv:2012.06866v1 fatcat:u4na2c45qzbkfjx3oedbwfvqmq

On generalized bent functions

Tor Helleseth, Alexander Kholosha
2010 2010 Information Theory and Applications Workshop (ITA)  
Bent functions were first introduced by Rothaus in 1976 as an interesting combinatorial object with the important property of having the maximum distance to all affine functions.  ...  This includes some recent constructions of weakly regular monomial and binomial bent functions and examples of non-weakly regular bent functions.  ...  Lemma 1: Take a permutation π : GF(2) n/2 → GF(2) n/2 and any mapping g : GF(2) n/2 → GF(2). Then f g,π (x, y) = x · π(y) + g(y) is a bent function.  ... 
doi:10.1109/ita.2010.5454124 dblp:conf/ita/HellesethK10 fatcat:npsg4xqxyvcwxndycngd4jqgv4

Completed GPS Covers All Bent Functions

Philippe Guillot
2001 Journal of combinatorial theory. Series A  
This proves that the completed class by composition with translations covers all the binary bent functions.  ...  In a recent paper, the so-called"generalized partial spread" (GPS) class which unifies almost all the known classes of bent functions is introduced.  ...  is that GPS does not cover the whole set of bent functions.  ... 
doi:10.1006/jcta.2000.3076 fatcat:vffck7ac5bdgbilzxdrri7pici

Self-dual Hadamard bent sequences [article]

Minjia Shi, Yaya Li, Wei Cheng, Dean Crnković, Denis Krotov, Patrick Solé
2022 arXiv   pre-print
We conjecture that if v is an even perfect square, a self-dual bent sequence of length v always exist.  ...  We introduce the strong automorphism group of Hadamard matrices, which acts on their associated self-dual bent sequences. We give an efficient algorithm to compute that group.  ...  We note that the concept of self-dual bent sequences being not invariant by Hadamard equivalence, classification of these become infeasible, even for matrices of small order.  ... 
arXiv:2203.16439v2 fatcat:dzjnsjnemfe3hmtfuqujzm64t4

New lower bounds on nonlinearity and a class of highly nonlinear functions [chapter]

Xian-Mo Zhang, Yuliang Zheng
1997 Lecture Notes in Computer Science  
Based on the study of these new lower bounds, we introduce a class of highly nonlinear Boolean functions with two di erences to bent function: (i) the dimension of these function is odd, (ii) they can  ...  Highly nonlinear Boolean functions occupy an important position in the design of secure block as well as stream ciphers.  ...  Part of the second author's work was completed while on sabbatical at the University of Tokyo. He would like to extend his thanks to Professor Hideki Imai for his hospitality.  ... 
doi:10.1007/bfb0027922 fatcat:nuzg5ijpxrei3kdet7n7ojd6ji

Cubic bent functions outside the completed Maiorana-McFarland class [article]

Alexandr Polujan, Alexander Pott
2020 arXiv   pre-print
In this paper we prove that in opposite to the cases of 6 and 8 variables, the Maiorana-McFarland construction does not describe the whole class of cubic bent functions in n variables for all n> 10.  ...  Moreover, we show that for almost all values of n, these functions can simultaneously be homogeneous and have no affine derivatives.  ...  Acknowledgments The authors would like to thank Pantelimon Stȃnicȃ for providing homogeneous cubic bent functions from [8, p. 149].  ... 
arXiv:1908.11271v3 fatcat:si2sqt4si5dvdg2ehdg4r5rex4

Generalised Bent Criteria for Boolean Functions (I) [article]

Constanza Riera, Matthew G. Parker
2005 arXiv   pre-print
Generalisations of the bent property of a boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms.  ...  The flat spectra of a quadratic boolean function are related to modified versions of its associated adjacency matrix.  ...  Theorem 8: There are no Completely I-bent boolean functions. Proof: Let s = (−1) p(x) . Let |R I | = n − 1. Then for U as defined in (14) , s ′ cannot be flat. D.  ... 
arXiv:cs/0502049v1 fatcat:mazmmfdiljb7bnoqgpqav4j6pq

Results on Bent Functions

Xiang-dong Hou, Philippe Langevin
1997 Journal of combinatorial theory. Series A  
Also see [9] by Hou on cubic bent functions. Despite extensive study, many questions about bent functions remain open.  ...  Starting with a single bent function, in a simple but very effective way, the construction produces a large number of new bent functions in the same number of variables.  ...  A RESTRICTION ON BENT FUNCTIONS Lemma 6.1. Let g, h # P m&1 .  ... 
doi:10.1006/jcta.1997.2804 fatcat:d42yzmuzfzc2ldwg43dpe3lwy4

Partial Spread and Vectorial Generalized Bent Functions [article]

Thor Martinsen, Wilfried Meidl, Pantelimon Stanica
2015 arXiv   pre-print
In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from _2^n to Z_2^t.  ...  Finally we point to a relation between vectorial gbent functions and relative difference sets.  ...  PS +/− gbent functions Being defined on a complete spread, for q = 2 with the construction in (1) we obtain P S − or complements of P Sbent functions.  ... 
arXiv:1511.01705v1 fatcat:w4o2fhvezzavlfbiwnbys6yyui

Partial spread and vectorial generalized bent functions

Thor Martinsen, Wilfried Meidl, Pantelimon Stănică
2016 Designs, Codes and Cryptography  
Explicitly, we describe gbent functions from F n 2 to Z 2 t , which can be seen as a gbent version of Dillon's P S ap class.  ...  Partial spread and vectorial generalized bent functions." Designs, Codes and Cryptography 85.1 (2017): 1-13.  ...  Acknowledgements Work by P. S. started during a very enjoyable visit at RICAM (Johann Radon Institute for Computational and Applied Mathematics), Austrian Academy of Sciences, in Linz, Austria.  ... 
doi:10.1007/s10623-016-0283-7 fatcat:tsw5jjfef5fcdnwrxfp62eh7wm

A huge class of infinite sequences of minimal binary linear codes with or without crossing the Ashikhmin-Barg's bound [article]

Fengrong Zhang, Enes Pasalic, René Rodríguez, Yongzhuang Wei
2020 IACR Cryptology ePrint Archive  
We first show that the use of a bent function g in the so-called direct sum of Boolean functions h(x, y) = f (x) + g(y), where f is arbitrary, induces minimal codes.  ...  g in s variables, which allows us to employ a suitable subspace of derivatives of g and generate minimal codes of dimension s + s/2 + 1 instead.  ...  In particular, when g ∈ B s is bent, the Walsh spectrum of the direct sum h(x, y) = f (x) + g(y) can be completely determined using the Walsh spectrum of f .  ... 
dblp:journals/iacr/ZhangPRW20 fatcat:7q3sdhubm5ejbout2nd75w4jly

Cubic bent functions

Xiang-dong Hou
1998 Discrete Mathematics  
We also determine all cubic bent functions in eight variables.  ...  If f is a cubic bent function in 2k variables, we show that when rs(f) <k, f is either obtained from a cubic bent function in 2k -2 variables, or is in a well-known family of bent functions.  ...  Bent functions have been studied extensively [l-5,7,8] . For k > 1, a bent function f E %k must have degree <k [8] . Quadratic bent functions are completely known.  ... 
doi:10.1016/s0012-365x(98)00008-9 fatcat:s2okk6k2g5genow2c64lmqma4q

A note of generalized bent functions

Ersan Akyildiz, Ismail Ş. Güloǧlu, Masatoshi /.Ikeda
1996 Journal of Pure and Applied Algebra  
In particular. the non-existence of Bent functions on L:". with p = 7 (mod 8) and r 2 1 is examined. The result obtained generalizes recent works of Bi (1991) and Pei (1993) .  ...  The main purpose of this paper is to understand the extent of the set of parameters for which no generalized bent functions exist.  ...  There is no generalized bent function on Z2Pr (r 2 1) for any prime p 3 7 mod 8 different front 7. Corollary 2. There is no generalized bent function Z2P,for any prime p = 7mod8.  ... 
doi:10.1016/0022-4049(94)00006-9 fatcat:m7uwv5zl6fbcfog2w2t5a5al6m

On Dillon's class H of bent functions, Niho bent functions and o-polynomials

Claude Carlet, Sihem Mesnager
2011 Journal of combinatorial theory. Series A  
All the elements of P S − have algebraic degree m exactly, but not all those of P S + (for instance, if m is even, then all quadratic functions are in P S + ). J.  ...  We first define the more general Partial Spreads class P S: it equals the union of P S − and P S + , where P S − (respectively, P S + ) is the set of all the sums (modulo 2) of the indicators of 2 m−1  ...  This completes the proof. 2 A similar property was observed by Maschietti in [19] (as recalled by Dobbertin in [10] ) for power functions.  ... 
doi:10.1016/j.jcta.2011.06.005 fatcat:spct5w47efdkzkeops7lilqnfa

Bent Functions of Maximal Degree

Ayça Cesmelioglu, Wilfried Meidl
2012 IEEE Transactions on Information Theory  
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.  ...  In this article a technique for constructing p-ary bent functions from plateaued functions is presented. This generalizes earlier techniques of constructing bent from nearbent functions.  ...  The function f is called a bent function if | f (b)| 2 = p n for all b ∈ V n .  ... 
doi:10.1109/tit.2011.2170053 fatcat:pnrjk7hifrdx7il3cj7qx4w76q
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