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We consider representations of an arbitrary permutation π of degree 2n, n ⩾ 3, by products of the so-called (2n)-permutations (any cycle of such a permutation has length 2). We show that any even permutation is represented by the product of four (2n)-permutations. Products of three (2n)-permutations cannot represent all even permutations. Any odd permutation is realized (for odd n) by a product of five (2n)-permutations. [ABSTRACT FROM AUTHOR]

The paper studies mappings F of binary vector spaces of large dimensions. The mappings are assumed to be represented by deep branching superpositions of local non-linear mappings of low-dimensional spaces. We propose and investigate the methods for construction of probabilistic linear and differential relations connecting the arguments and the values of a mapping F. Relation selection is based on optimization not the exact probability of satisfying these relations, but some approximation of it since it is easier to estimate. We prove theorems on exact values of the probability of satisfying the relations obtained, identify the shortcomings and features of the proposed approach to the relation construction and illustrate them by a number of examples. We discuss the role of the developed theory for cryptographic synthesis. [ABSTRACT FROM AUTHOR]

We give an elementary proof that by multiplication on left and right by affine permutations A, B ∈ AGL(3, 2) each permutation π : GF(2)3 → GF(2)3 may be reduced to one of the 4 permutations for which the 3 × 3-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory. [ABSTRACT FROM AUTHOR]

We give an elementary proof that by multiplication on left and right by affine permutations A, B ∈ AGL(3, 2) each permutation π : GF(2)3 → GF(2)3 may be reduced to one of the 4 permutations for which the 3 × 3-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory. [ABSTRACT FROM AUTHOR]