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1.
In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6. 相似文献
2.
Xiang-dong Hou 《Proceedings of the American Mathematical Society》2000,128(4):987-996
We prove a 2-adic inequality for the coefficients of binary bent functions in their polynomial representations. The 2-adic inequality implies a family of identities satisfied by the coefficients. The identities also lead to the discovery of some new affine invariants of Boolean functions on .
3.
Bent functions are those Boolean functions whose Hamming distance to the Reed-Muller code of order 1 equal 2n-1-2n/2-1 (where the number n of variables is even). These combinatorial objects, with fascinating properties, are rare. Few constructions are known, and it is difficult to know whether the bent functions they produce are peculiar or not, since no way of generating at random bent functions on 8 variables or more is known.The class of bent functions contains a subclass of functions whose properties are still stronger and whose elements are still rarer. Youssef and Gong have proved the existence of such hyper-bent functions, for every even n. We prove that the hyper-bent functions they exhibit are exactly those elements of the well-known PSap class, introduced by Dillon, up to the linear transformations x?δx, . Hyper-bent functions seem still more difficult to generate at random than bent functions; however, by showing that they all can be obtained from some codewords of an extended cyclic code Hn with small dimension, we can enumerate them for up to 10 variables. We study the non-zeroes of Hn and we deduce that the algebraic degree of hyper-bent functions is n/2. We also prove that the functions of class PSap are some codewords of weight 2n-1-2n/2-1 of a subcode of Hn and we deduce that for some n, depending on the factorization of 2n-1, the only hyper-bent functions on n variables are the elements of the class , obtained from PSap by composing the functions by the transformations x?δx, δ≠0, and by adding constant functions. We prove that non- hyper-bent functions exist for n=4, but it is not clear whether they exist for greater n. We also construct potentially new bent functions for n=12. 相似文献
4.
The question if there exist nonnormal bent functions was an open question for several years. A Boolean function in n variables is called normal if there exists an affine subspace of dimension n/2 on which the function is constant. In this paper we give the first nonnormal bent function and even an example for a nonweakly normal bent function. These examples belong to a class of bent functions found in [J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters, in: Finite Fields and Applications, to appear], namely the Kasami functions. We furthermore give a construction which extends these examples to higher dimensions. Additionally, we present a very efficient algorithm that was used to verify the nonnormality of these functions. 相似文献
5.
Hans Dobbertin Anne Canteaut Patrick Felke 《Journal of Combinatorial Theory, Series A》2006,113(5):779-798
A Boolean function with an even number n=2k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f(x)=tr(α1xd1+α2xd2), α1,α2,x∈F2n, are considered, where the exponents di (i=1,2) are of Niho type, i.e. the restriction of xdi on F2k is linear. We prove for several pairs of (d1,d2) that f is a bent function, when α1 and α2 fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F2n are bijective. 相似文献
6.
The problem of computing the automorphism groups of an elementary Abelian Hadamard difference set or equivalently of a bent function seems to have attracted not much interest so far. We describe some series of such sets and compute their automorphism group. For some of these sets the construction is based on the nonvanishing of the degree 1-cohomology of certain Chevalley groups in characteristic two. We also classify bent functions f such that Aut(f) together with the translations from the underlying vector space induce a rank 3 group of automorphisms of the associated symmetric design. Finally, we discuss computational aspects associated with such questions. 相似文献
7.
This paper discusses a kind of bent functions that have some symmetric properties about some variables. Section 2 mainly discusses the bent functions symmetric about some two variables and gives the necessary and sufficient condition for these functions. Section 3 gives algebraic expressions of some bent functions. 相似文献
8.
Construction of bent functions from near-bent functions 总被引:1,自引:0,他引:1
Gregor Leander 《Journal of Combinatorial Theory, Series A》2009,116(4):960-970
We give a construction of bent functions in dimension 2m from near-bent functions in dimension 2m−1. In particular, we give the first ever examples of non-weakly-normal bent functions in dimensions 10 and 12, which demonstrates the significance of our construction. 相似文献
9.
An Erratum has been published for this article in Journal of Combinatorial Designs 14: 82–82, 2006 . We give the equivalence between perfect nonlinear functions and appropriate splitting semi‐regular relative difference sets, construct a class of splitting relative difference sets by using Galois rings and bent functions, and prove that there exists a 4‐phase perfect nonlinear function if and only if the number of input variables is at least twice the number of output variables. © 2005 Wiley Periodicals, Inc. 相似文献
10.
11.
Coulter–Matthews (CM) bent functions are from to defined by , where and (α,2n)=1. It is not known if these bent functions are weakly regular in general. In this paper, we show that when n is even and α=n+1 (or n−1), the CM bent function is weakly regular. Moreover, we explicitly determine the dual of the CM bent function in this case. The dual is a bent function not reported previously. 相似文献
12.
In this paper, we present partial results towards the conjectured nonexistence of homogeneous rotation symmetric bent functions having degree > 2. 相似文献
13.
In this article a technique for constructing p-ary bent functions from near-bent functions is presented. This technique is then used to obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions. 相似文献
14.
G. G. Amanzhaev 《Mathematical Notes》1998,64(5):557-561
For discrete analogs of classes of functions of finite smoothness, we study the quantity log Approx characterizing the minimal
necessary length of tables that allow us to reconstruct functions from these classes with error not exceeding 1 in the metric
of the spaceL
p
.
Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 643–647, November, 1998.
The author wishes to express his gratitude to his scientific adviser O. B. Lupanov for setting the problem and attention to
the author's work.
This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01068. 相似文献
15.
p-Ary and q-ary versions of certain results about bent functions and resilient functions 总被引:2,自引:1,他引:1
Using the Teichmüller character and Gauss sums, we obtain the following results concerning p-ary bent functions and q-ary resilient functions: (1) a characterization of certain q-ary resilient functions in terms of their coefficients; (2) stronger upper bounds for the degree of p-ary bent functions; (3) determination of all bent functions on
; (4) a characterization of ternary weakly regular bent functions in terms of their coefficients. 相似文献
16.
In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth–Kumar–Martinsen (HKM) difference set and the Lin difference set can be easily obtained. Also, the p-rank of a Singer difference set is reviewed and the characteristic polynomial is calculated using our approach. 相似文献
17.
Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence. 相似文献
18.
Alexander Pott 《Discrete Mathematics》2008,308(13):2854-2861
The well-known difference sets have various connections with sequences and their correlation properties. It is the purpose of this note to give two more applications of the (not so well known) relative difference sets: we use them to construct difference triangles (based on an idea of A. Ling) and we show that a certain nonexistence result for semiregular relative difference sets implies the nonexistence of negaperiodic autocorrelation sequences (answering a question of Parker [Even length binary sequence families with low negaperiodic autocorrelation, in: Applied Algebra, Algebraic Algorithms and Error-correcting Codes, Melbourne, 2001, Lecture Notes in Computer Science, vol. 2227, Springer, Berlin, 2001, pp. 200-209.]). 相似文献
19.
It is established that a finite p-solvable group presenting the product of two of its Schmidt subgroups has p-length at most 2. 相似文献
20.
Hisayasu Kurata 《Discrete Applied Mathematics》2008,156(1):103-109
We discuss the p-harmonicity of the linear combination of p-harmonic functions in the Euclidean space and on a tree. If p≠2, the p-harmonicity is non-linear, i.e., the linear combination of p-harmonic functions need not be p-harmonic. In spite of this non-linear nature, we find some p-harmonic functions whose linear combinations become p-harmonic. 相似文献