Volume 26, Issue 4

The paper is concerned with combinatorial description of almost perfect nonlinear functions (APN-functions). A complete characterization of n -place APN-functions in terms of ( n − 1)-place subfunctions is obtained. An n -place function is shown to be an APN-function if and only if each of its ( n − 1)-place subfunctions is either an APN-function or has the differential uniformity 4 and the admissibility conditions hold. A detailed characterization of 2, 3 or 4-place APN-functions is presented.

The distribution function of the minimum distance (the minimum weight of nonzero codewords) of a random linear code over a finite field is studied. Expicit bounds in the form of inequalities and asymptotic estimates for this distribution function are obtained.

In a sequence of independent positive random variables with the same continuous distribution function a monotonic subsequence of record values is chosen. A corresponding sequence of record times divides the initial sequence into interrecord intervals. Let αij (i⩾1,j=1,…,i)$\alpha_i^j \ (i\geqslant 1, \, j = 1, \ldots, i)$ be the number of random variables in the interval between i -th and ( i + 1)-th record moments with values between ( j − 1)-th and j -th records. Explicit formulas for the joint distributions of the random variables αij,1⩽j⩽i⩽n$\alpha_i^j,\,1\leqslant j\leqslant i\leqslant n$, are derived, limit theorems for the distributions of αij$\alpha_i^j$ for i − j → ∞ are proved.

We study clones (closed sets of operations that contain projections) and superclones on finite sets. According to A. I. Mal’tsev a clone may be considered as an algebra. If we replace algebra universe with a set of multioperations and add the operation of simplest equation solvability then we will obtain an algebra called a superclone. The paper establishes Galois connection between clones and superclones.

The additive groups of the residue ring ℤ 2 n and of the vector space V n over the field GF (2), as well as the group G n generated by these additive groups, share common imprimitivity systems and enter as subgroups into the Sylow 2-subgroup of the symmetric group S (ℤ 2 n ). These groups are used in cryptography as an encryption tool with the operations of addition in V n and ℤ 2 n . The permutation structure of the subgroups of the group G n is presented. The kernels of homomorphisms which correspond to various systems of imprimitivity, the normal subgroups, and some modular representations of the group G n over the field GF (2) are described.