One approach to constructing a multiply transitive class of block transformations

Let П be an arbitrary finite set, B(Q) - the collection of all binary operations defined on the set П, B*(Q) - the family of all binary operations that are invertible in the right variable, x₁,... ,x_n - variables over П, and *₁,..., *_k - general symbols of binary operations. A fixed cortege W = (w₁,..., w_m) of formulas in the alphabet {x₁,..., x_n, *₁,..., } implements the mapping W^Fl'-"'^Ffc: П^га ^ when replacing symbols *₁,..., with an arbitrary binary operations F₁,..., е В(II), respectively. In this paper we offer a visual representation of the transformation family {W^Fl>">^Ffc : F₁,..., е B*(II)} in the form of a binary functional network. This representation allows us to strictly describe the methods of research on the multiply transitivity of an arbitrary family {W: F₁,..., е В*(II)}. In addition, network view makes it possible to construct cortege of formulas W = (w₁,... ,w_n) such that the family {W^Fl''"'^Ffc : F₁,...,F_k е B*(II)} is multiply transitive. Moreover, some block ciphers (Blowfish, Twofish, etc), in which the S-boxes depend on the key, can be "approximated" by family of the form {W: F₁,..., е В*(II)} and, as a result, it becomes possible to evaluate the multiple transitivity of such ciphers.

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