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JOURNALS // Prikladnaya Diskretnaya Matematika. Supplement // Archive

Prikl. Diskr. Mat. Suppl., 2020 Issue 13, Pages 6–8 (Mi pdma481)

This article is cited in 1 paper

Theoretical Foundations of Applied Discrete Mathematics

Refractive bijections in Steiner triples

M. V. Vedunova, K. L. Geut, A. O. Ignatova, S. S. Titov

Urals State University of Railway Transport, Ekaterinburg

Abstract: The paper deals with refractive bijections in Steiner triples used in the construction of matroids and secret sharing schemes. Refractors are understood to mean mappings $F$ of a quasigroup into itself satisfying the condition $F (x * y) \neq F (x) * F (y)$ for any $x \neq y$. The necessary conditions for the existence of APN-bijections in $\mathrm{GF}(2^n)$ are found, for $N=7$ the superposition of any two refractive bijections is not refractive. It is found that for $N=9$, $13$ and $2^n-1$ elements for odd $n$ not divisible by three, there are three Steiner triples systems without common triples. Refractive bijections are proposed for systems of Steiner triples without common triples for $N=13$. A counterexample is obtained to the hypothesis that each homogeneous matroid defines a certain block scheme using sets of refractive bijections, for $N=7$ such $S, S', S''$ do not exist. Functions that are APN-bijections are given. The condition allowing to construct homogeneous matroids that are not reduced to block scheme used in secret sharing schemes using Steiner linear triples systems is revealed, and a refractive bijection that is not an APN-function is also found, for instance $F(x)=x^{-3}$.

Keywords: refracting bijections, Steiner quasigroups, matroids.

UDC: 519.151, 519.725, 519.165

DOI: 10.17223/2226308X/13/1



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