Anbar Meidl, Nurdagül and Kalaycı, Tekgül and Meidl, Wilfried (2021) Analysis of (n, n)functions obtained from the MaioranaMcFarland class. IEEE Transactions on Information Theory, 67 (7). pp. 48914901. ISSN 00189448 (Print) 15579654 (Online)
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Official URL: http://dx.doi.org/10.1109/TIT.2021.3079223
Abstract
Pott et al. (2018) showed that F(x) = x2r Trn m(x), n = 2m, r ≥ 1, is a nontrivial example of a vectorial function with the maximal possible number 2n 2m of bent components. Mesnager et al. (2019) generalized this result by showing conditions on Λ(x) = x+ ∑σ j=1 αjx2tj, αj ∈ 2 F2m, under which F(x) = x2r Trn m(Λ(x)) has the maximal possible number of bent components. We simplify these conditions and further analyse this class of functions. For all related vectorial bent functions F(x) = Trn m(γF(x)), γ ∈ 2 F2n F2m, which as we will point out belong to the MaioranaMcFarland class, we describe the collection of the solution spaces for the linear equations DaF(x) = F(x) + F(x + a) + F(a) = 0, which forms a spread of F2n. Analysing these spreads, we can infer neat conditions for functions H(x) = (F(x);G(x)) from F2n to F2m × F2m to exhibit small differential uniformity (for instance for Λ(x) = x and r = 0 this fact is used in the construction of Carlet’s, PottZhou’s, Taniguchi’s APNfunction). For some classes of H(x) we determine differential uniformity and with a method based on Bezout’s theorem nonlineariy.
Item Type:  Article 

Subjects:  Q Science > QA Mathematics > QA150272.5 Algebra Q Science > QA Mathematics 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Nurdagül Anbar Meidl 
Date Deposited:  22 Jun 2021 16:01 
Last Modified:  30 Aug 2022 23:30 
URI:  https://research.sabanciuniv.edu/id/eprint/41561 
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Analysis of (n, n)functions obtained from the MaioranaMcFarland class. (deposited 25 May 2021 16:42)
 Analysis of (n, n)functions obtained from the MaioranaMcFarland class. (deposited 22 Jun 2021 16:01) [Currently Displayed]