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  • On the maximum number of bent components of vectorial functions
    Pott, Alexander ...
    In this paper, we show that the maximum number of bent component functions of a vectorial function ▫$F:GF(2)^{n}\to GF(2)^{n}$▫ is ▫$2^{n}-2^{n/2}$▫. We also show that it is very easy to construct ... such functions. However, it is a much more challenging task to find such functions in polynomial form ▫$F\in GF(2^{n})[x]$▫, where ▫$F$▫ has only a few terms. The only known power functions having such a large number of bent components are ▫$x^d$▫, where ▫$d=2^{n/2}+1$▫. In this paper, we show that the binomials ▫$F^{i}(x) = x^{2^{i}}(x+x^{2^{n/2}})$▫ also have such a large number of bent components, and these binomials are inequivalent to the monomials ▫$x^{2^{n/2}+1}$▫ if ▫$0<i<n/2$▫. In addition, the functions ▫$F^i$▫ have differential properties much better than ▫$x^{2^{n/2}+1}$▫. We also determine the complete Walsh spectrum of our functions when ▫$n/2$▫ is odd and ▫$\mathrm{gcd} (i, n/2)=1$▫.
    Source: IEEE transactions on information theory. - ISSN 0018-9448 (Vol. 64, no. 1, Jan. 2018, str. 403-411)
    Type of material - article, component part ; adult, serious
    Publish date - 2018
    Language - english
    COBISS.SI-ID - 31065383

    Link(s):

    http://dx.doi.org/10.1109/TIT.2017.2749421
    DOI

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