The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups

The limited deficit method is described, which allows constructing new orthomorphisms (almost orthomorphisms) of groups with the use of those already known. A class of transformations is described under which the set of all orthomorphisms (almost orthomorphisms) remains invariant. It is conjectured that the set of all orthomorphisms (almost orthomorphisms) is generated by transformations implemented by the limited deficit method. This conjecture is verified for all Abelian groups of order at most 12. The spectral-linear method and the spectral-differential method of design of permutations over the additive group of the field ${\rm{\mathbb F}}_{2^{m}}$ ($m=4,\ldots,8$) are used to construct orthomorphisms with sufficiently high values of the most important cryptographic parameters.