667 research outputs found

    A Further Study of Vectorial Dual-Bent Functions

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    Vectorial dual-bent functions have recently attracted some researchers' interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions and linear codes. In this paper, we further study vectorial dual-bent functions F:Vn(p)→Vm(p)F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}, where 2≤m≤n22\leq m \leq \frac{n}{2}, Vn(p)V_{n}^{(p)} denotes an nn-dimensional vector space over the prime field Fp\mathbb{F}_{p}. We give new characterizations of certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A) in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. When p=2p=2, we characterize vectorial dual-bent functions with Condition A in terms of bent partitions. Furthermore, we characterize certain bent partitions in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. For general vectorial dual-bent functions F:Vn(p)→Vm(p)F: V_{n}^{(p)}\rightarrow V_{m}^{(p)} with F(0)=0,F(x)=F(−x)F(0)=0, F(x)=F(-x) and 2≤m≤n22\leq m \leq \frac{n}{2}, we give a necessary and sufficient condition on constructing association schemes. Based on such a result, more association schemes are constructed from vectorial dual-bent functions

    Generalized bent Boolean functions and strongly regular Cayley graphs

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    In this paper we define the (edge-weighted) Cayley graph associated to a generalized Boolean function, introduce a notion of strong regularity and give several of its properties. We show some connections between this concept and generalized bent functions (gbent), that is, functions with flat Walsh-Hadamard spectrum. In particular, we find a complete characterization of quartic gbent functions in terms of the strong regularity of their associated Cayley graph.Comment: 13 pages, 2 figure

    Decomposing generalized bent and hyperbent functions

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    In this paper we introduce generalized hyperbent functions from F2nF_{2^n} to Z2kZ_{2^k}, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions from F2nF_{2^n} to Z2kZ_{2^k} consist of components which are generalized (hyper)bent functions from F2nF_{2^n} to Z2k′Z_{2^{k^\prime}} for some k′<kk^\prime < k. For odd nn, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even nn, where the associated Boolean functions are bent.Comment: 24 page

    Landscape Boolean Functions

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    In this paper we define a class of Boolean and generalized Boolean functions defined on F2n\mathbb{F}_2^n with values in Zq\mathbb{Z}_q (mostly, we consider q=2kq=2^k), which we call landscape functions (whose class containing generalized bent, semibent, and plateaued) and find their complete characterization in terms of their components. In particular, we show that the previously published characterizations of generalized bent and plateaued Boolean functions are in fact particular cases of this more general setting. Furthermore, we provide an inductive construction of landscape functions, having any number of nonzero Walsh-Hadamard coefficients. We also completely characterize generalized plateaued functions in terms of the second derivatives and fourth moments.Comment: 19 page

    Value Distributions of Perfect Nonlinear Functions

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    In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.Comment: 28 pages. minor revisions of the previous version. The paper is now identical to the published version, outside of formattin
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