About nonlinear orthomorphisms and total substitutions

Abstract

: The work is dedicated to the study of substitutions, orthomorphisms of quasigroups and abelian groups, that found their application in cryptography, encryption, authentication and other areas of information security. The concepts of complete substitutions, nonlinear orthomorphisms, their properties and parameters have been highlighted. Assertions about the dependence of the properties of permutations on the composition of their spectra have been established. Constructions and examples of orthomorphisms have been proposed, including through the division of quasigroups into subquasigroups. Spectral properties of permutations have been formulated, which characterize permutations of O (2)- strong ciphers. Examples have been given to show the established assertions and conclusions. The connection of the results obtained with the Laia-Messi protocol, Steiner triples, and the Kirkman problem about schoolgirls has been discussed. The issue of cryptographic interpretation of fixed points under a shift of substitutions has been considered. The proposed methods and interpretations can be applied for theoretical generalizations and practical applications in information security.