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

2020
*
arXiv
*
pre-print

problem for (n,m)-

arXiv:2012.06866v1
fatcat:u4na2c45qzbkfjx3oedbwfvqmq
*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. ...##
###
On generalized bent functions

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*. ...

##
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Completed GPS Covers All Bent Functions

2001
*
Journal of combinatorial theory. Series A
*

This proves that the

doi:10.1006/jcta.2000.3076
fatcat:vffck7ac5bdgbilzxdrri7pici
*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*. ...##
###
Self-dual Hadamard bent sequences
[article]

2022
*
arXiv
*
pre-print

We conjecture that if v is an even perfect square, a self-dual

arXiv:2203.16439v2
fatcat:dzjnsjnemfe3hmtfuqujzm64t4
*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. ...##
###
New lower bounds on nonlinearity and a class of highly nonlinear functions
[chapter]

1997
*
Lecture Notes in Computer Science
*

Based on the study of these new lower bounds, we introduce a class of highly nonlinear Boolean

doi:10.1007/bfb0027922
fatcat:nuzg5ijpxrei3kdet7n7ojd6ji
*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. ...##
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Cubic bent functions outside the completed Maiorana-McFarland class
[article]

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

arXiv:1908.11271v3
fatcat:si2sqt4si5dvdg2ehdg4r5rex4
*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]. ...##
###
Generalised Bent Criteria for Boolean Functions (I)
[article]

2005
*
arXiv
*
pre-print

Generalisations of the

arXiv:cs/0502049v1
fatcat:mazmmfdiljb7bnoqgpqav4j6pq
*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. ...##
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Results on Bent Functions

1997
*
Journal of combinatorial theory. Series A
*

Also see [9] by Hou on cubic

doi:10.1006/jcta.1997.2804
fatcat:d42yzmuzfzc2ldwg43dpe3lwy4
*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 . ...##
###
Partial Spread and Vectorial Generalized Bent Functions
[article]

2015
*
arXiv
*
pre-print

In this paper we generalize the partial spread class and

arXiv:1511.01705v1
fatcat:w4o2fhvezzavlfbiwnbys6yyui
*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**S*−*bent**functions*. ...##
###
Partial spread and vectorial generalized bent functions

2016
*
Designs, Codes and Cryptography
*

Explicitly, we describe gbent

doi:10.1007/s10623-016-0283-7
fatcat:tsw5jjfef5fcdnwrxfp62eh7wm
*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. ...##
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A huge class of infinite sequences of minimal binary linear codes with or without crossing the Ashikhmin-Barg's bound
[article]

2020
*
IACR Cryptology ePrint Archive
*

We first show that the use of a

dblp:journals/iacr/ZhangPRW20
fatcat:7q3sdhubm5ejbout2nd75w4jly
*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 . ...##
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Cubic bent functions

1998
*
Discrete Mathematics
*

We also determine

doi:10.1016/s0012-365x(98)00008-9
fatcat:s2okk6k2g5genow2c64lmqma4q
*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. ...##
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A note of generalized bent functions

1996
*
Journal of Pure and Applied Algebra
*

In particular. the non-existence of

doi:10.1016/0022-4049(94)00006-9
fatcat:m7uwv5zl6fbcfog2w2t5a5al6m
*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. ...##
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On Dillon's class H of bent functions, Niho bent functions and o-polynomials

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*. ...

##
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Bent Functions of Maximal Degree

2012
*
IEEE Transactions on Information Theory
*

Applying the construction of

doi:10.1109/tit.2011.2170053
fatcat:pnrjk7hifrdx7il3cj7qx4w76q
*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 . ...
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