Refractive bijections in Steiner triples

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) = F(x) * F(y) for any x = y. The necessary conditions for the existence of APN-bijections in GF(2ⁿ) are found, for N = 7 the superposition of any two refractive bijections is not refractive. It is found that for N = 9, 13 and 2ⁿ - 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.

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