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 Prikl. Diskr. Mat. Suppl., 2019, Issue 12, Pages 75–77 (Mi pdma438)

Discrete Functions

Class of Boolean functions constructed using significant bits of linear recurrences over the ring $\mathbb{Z}_{2^n}$

D. H. Hernández Piloto

LLC "Certification Research Center", Moscow

Abstract: In this paper, we study a class of functions built with the help of significant bits sequences on the ring $\mathbb{Z}_{2 ^ n}$. This class is built with the use of a function $\psi: \mathbb{Z}_{2 ^ n} \rightarrow \mathbb{Z}_2$. In public literature, there are results for a linear function $\psi$. Here, we use a non-linear $\psi$ function for this set. The period of a polynomial $F$ in the ring $\mathbb{Z}_{2^n}$ is equal to $T(F \bmod 2)2^{\alpha}$, where $\alpha \in \{0,\ldots, n-1\}$. The polynomials for which $T(F) = T(F \bmod 2)$, i.e. $\alpha = 0$, are called marked polynomials. For our class, we use a marked polynomial of the maximum period. We show the bounds of the given class: non-linearity, the weight of the functions, the Hamming distance between functions. The Hamming distance between these functions and functions of other known classes is also given.

Keywords: Boolean functions, linear recurrent sequences, significant bits sequences.

DOI: https://doi.org/10.17223/2226308X/12/23

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UDC: 621.391:519.7

Citation: D. H. Hernández Piloto, “Class of Boolean functions constructed using significant bits of linear recurrences over the ring $\mathbb{Z}_{2^n}$”, Prikl. Diskr. Mat. Suppl., 2019, no. 12, 75–77

Citation in format AMSBIB
\Bibitem{Her19} \by D.~H.~Hern\'andez Piloto \paper Class of Boolean functions constructed using significant bits of linear recurrences over the ring $\mathbb{Z}_{2^n}$ \jour Prikl. Diskr. Mat. Suppl. \yr 2019 \issue 12 \pages 75--77 \mathnet{http://mi.mathnet.ru/pdma438} \crossref{https://doi.org/10.17223/2226308X/12/23}