Refractive bijections in Steiner triples

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

Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 6-8

Résumé

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 $F(x)=x^{-3}$.

Détail

Keywords: refracting bijections, Steiner quasigroups, matroids.