Yazar "Meidl, Wilfried" seçeneğine göre listele

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    Bent and vectorial bent functions, partial difference sets, and strongly regular graphs
    (Amer Inst Mathematical Sciences-Aims, 2018) Çesmelioğlu, Ayça; Meidl, Wilfried
    Bent and vectorial bent functions have applications in cryptography and coding and are closely related to objects in combinatorics and finite geometry, like difference sets, relative difference sets, designs and divisible designs. Bent functions with certain additional properties yield partial difference sets of which the Cayley graphs are always strongly regular. In this article we continue research on connections between bent functions and partial difference sets respectively strongly regular graphs. For the first time we investigate relations between vectorial bent functions and partial difference sets. Remarkably, properties of the set of the duals of the components play here an important role. Seeing conventional bent functions as 1-dimensional vectorial bent functions, some earlier results on strongly regular graphs from bent functions follow from our more general results. Finally we describe a recursive construction of infinitely many partial difference sets with a secondary construction of p-ary bent functions.
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    Bent functions, spreads, and o-polynomials
    (Siam Publications, 2015) Çeşmelioğlu, Ayça; Meidl, Wilfried; Pott, Alexander
    We show that bent functions f from F(p)m x F(p)m to F-p, which are constant or affine on the elements of a given spread of F(p)m x F(p)m, either arise from partial spread bent functions, or they are Boolean and a generalization of Dillon's class H. For spreads of a presemifield S, we show that a bent function of the second class corresponds to an o-polynomial of a presemifield in the Knuth orbit of S. In contrast to the finite fields case, we have to consider pairs of (pre) semifields in a Knuth orbit. We give a canonical example of an o-polynomial for commutative presemifields (which also defines a hyperoval on the semifield plane) and show that the corresponding bent functions belong to the completed Maiorana-McFarland class. Using Albert's twisted fields and Kantor's family of presemifields, we explicitly present examples of such bent functions.
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    Öğe
    Non weakly regular bent polynomials from vectorial quadratic functions
    (Amer Mathematical Soc, 2015) Çeşmelioğlu, Ayça; Meidl, Wilfried
    The Fourier spectrum of some classes of vectorial quadratic functions is analysed. We obtain counting functions for vectorial quadratic functions with prescribed Fourier spectrum using discrete Fourier transform. Vectorial quadratic functions with prescribed Fourier spectrum are then used to construct polynomials in F-pn [x] inducing non weakly regular bent functions from F-pn. to F-p in even dimension n. With a class of vectorial 2-plateaued binomials we explicitly present one infinite class of non weakly regular bent functions with a simple representation.
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    Partially bent functions and their properties
    (Cambridge Univ Press, 2014) Çeşmelioğlu, Ayça; Meidl, Wilfried; Topuzoğlu, Alev
    A function f : F-p(n) -> F-p is called partially bent if for all a is an element of F-p(n) the derivative D(a)f(x) = f(x + a) - f(x) is constant or balanced, i.e., every value in F-p is taken on p(n-1) times. Bent functions have balanced derivatives D(a)f for all nonzero a is an element of F-p(n), hence are partially bent. Partially bent functions may be balanced and highly nonlinear, and thus have favorable properties for cryptographic applications in stream and block ciphers. Hence they are of independent interest. Partially bent functions are also used to construct new bent functions. The aim of this article is to provide a deeper understanding of partially bent functions. We collect their properties and describe partially bent functions with appropriate generalizations of relative difference sets and difference sets. The descriptions of bent functions as relative difference sets and of Hadamard difference sets in characteristic 2, follow from our result as special cases. We describe Hermitian matrices related to partially bent functions and interpret a secondary construction of bent functions from partially bent functions in terms of relative difference sets.
  • Küçük Resim
    Öğe
    There are infinitely many bent functions for which the dual is not bent
    (Ieee-Inst Electrical Electronics Engineers Inc, 2016) Ceşmelioğlu, Ayça; Meidl, Wilfried; Pott, Alexander
    Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs, since their duals are also bent functions. In general, this does not apply to non-weakly regular bent functions. However, the first known construction of non-weakly regular bent functions by Cesmelioglu et al. yields bent functions for which the dual is also bent. In this paper, the first construction of non-weakly regular bent functions for which the dual is not bent is presented. We call such functions nondual-bent functions. Until now, only sporadic examples found via computer search were known. We then show that with the direct sum of bent functions and with the construction by Cesmelioglu et al., one can obtain infinitely many non-dual-bent functions once one example of a non-dual-bent function is known.
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    Vectorial bent functions and their duals
    (Elsevier Science Inc, 2018) Çesmelioğlu, Ayça; Meidl, Wilfried; Pott, Alexander
    Motivated by the observation that for two (weakly regular) bent functions f, g for which also f + g is bent, the sum f* + g* of their duals f and g* is sometimes but not always bent, we initiate the study of duality for vectorial bent functions. We propose and investigate two concepts of self-duality for vectorial bent functions, self-duality and weak self-duality. (C) 2018 Elsevier Inc. All rights reserved.