One approach to constructing a multiply transitive class of block transformations | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2018. № 42. DOI: 10.17223/20710410/42/2

In this work, we continue to study the cryptographic properties of block transformations of a new type, which can be used to construct hash functions and block ciphers. Let Q be an arbitrary finite set, q(Q) be the collection of all binary quasigroups defined on the set Q, and £^F: Qⁿ ^ Qⁿ be a mapping that is implemented by a network £ of width n with one binary operation F g q(Q). The network £ is called bijective if the mapping £^f is bijective for each F g q(Q) and all finite sets Q. The networks £i, £₂ are called equivalent if the map £^F of £₁ coincides with the map £^F of £₂ for each F g q(Q) and for all finite sets Q. It is not difficult to define the elementary networks by analogy with the elementary matrices and prove that every bijective network £ is equivalent to a unique product of elementary networks. This product is called the canonical representation of £ and its length is denoted by ||£||. A bijective network £ is called k-transitive for Q if the family {£^F : F g q(Q)} is k-transitive. We prove that the bijective network £ is k-transitive for all sufficiently large finite sets iff £ is k-transitive for some finite set Q such that |Q| ^ k||£||+kn. In addition, we propose an effective method for verifying the network's k-transitivity for all sufficiently large finite sets, namely, the bijective network £ is k-transitive for Q such that |Q| ^ k||£|| + kn whenever it is k-transitive for some (k + 1)-element subset of Q. Also, we describe an algorithm for constructing k-transitive networks. For a given bijective network £ of a width n, the algorithm adds 6n - 7 elementary networks to the canonical representation of £ without changing the existing contents. As a result of these modifications, we obtain a bijective network £ that is k-transitive for every sufficiently large finite set Q, namely for |Q| ^ k||£|| + kn.