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    Öğe
    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.
  • 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.
  • [ X ]
    Öğe
    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.