13,077 research outputs found

    On the normality of pp-ary bent functions

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    Depending on the parity of nn and the regularity of a bent function ff from Fpn\mathbb F_p^n to Fp\mathbb F_p, ff can be affine on a subspace of dimension at most n/2n/2, (n1)/2(n-1)/2 or n/21n/2- 1. We point out that many pp-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) n/2n/2-normal, i.e. affine on a n/2n/2-dimensional subspace. However applying an algorithm by Canteaut et.al., some Boolean bent functions were shown to be not n/2n/2- normal. We develop an algorithm for testing normality for functions from Fpn\mathbb F_p^n to Fp\mathbb F_p. Applying the algorithm, for some bent functions in small dimension we show that they do not take on the bound on normality. Applying direct sum of functions this yields bent functions with this property in infinitely many dimensions.Comment: 13 page

    Semifields, relative difference sets, and bent functions

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    Recently, the interest in semifields has increased due to the discovery of several new families and progress in the classification problem. Commutative semifields play an important role since they are equivalent to certain planar functions (in the case of odd characteristic) and to modified planar functions in even characteristic. Similarly, commutative semifields are equivalent to relative difference sets. The goal of this survey is to describe the connection between these concepts. Moreover, we shall discuss power mappings that are planar and consider component functions of planar mappings, which may be also viewed as projections of relative difference sets. It turns out that the component functions in the even characteristic case are related to negabent functions as well as to Z4\mathbb{Z}_4-valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite fields", 09-13 December 2013, Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

    Scalar fields, bent branes, and RG flow

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    This work deals with braneworld scenarios driven by real scalar fields with standard dynamics. We show how the first-order formalism which exists in the case of four dimensional Minkowski space-time can be extended to de Sitter or anti-de Sitter geometry in the presence of several real scalar fields. We illustrate the results with some examples, and we take advantage of our findings to investigate renormalization group flow. We have found symmetric brane solutions with four-dimensional anti-de Sitter geometry whose holographically dual field theory exhibits a weakly coupled regime at high energy.Comment: 22 pages, 7 figure
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