Volume 28, Issue 4

The average-case complexity of computation of underdetermined functions by straight-line programs with conditional stop over the basis of all at most two-place Boolean functions is considered. Correct order estimates of the average-case complexity of functions with maximum average-case complexity among all underdetermined functions are derived depending on the degree of their determinacy, the size of their domain, and the size of their support.

For a scheme of allocation of distinguishable particles into indistinguishable cells we describe types of representation, numbering and enumerating of its outcomes in terms of the transition graph; this graph allows, in particular, to find the distribution on the set of outcomes. Several methods of statistical simulation of scheme outcomes are described.

Groups which are most frequently used as key addition groups in iterative block ciphers include the regular permutation representation Vn+$\begin{array}{} \displaystyle V_{n}^{+} \end{array}$ of the group of vector key addition, the regular permutation representation Z2n+$\begin{array}{} \displaystyle \mathbb{Z}_{{2^n}}^{+} \end{array}$ of the additive group of the residue ring, and the regular permutation representation Z2n+1⊙$\begin{array}{} \displaystyle \mathbb{Z}_{{2^n} + 1}^ \odot \end{array}$ of the multiplicative group of a prime field (in the case where 2 n + 1 is a prime number). In this work we consider the extension of the group G n generated by Vn+$\begin{array}{} \displaystyle V_{n}^{+} \end{array}$ and Z2n+$\begin{array}{} \displaystyle \mathbb{Z}_{{2^n}}^{+} \end{array}$ by means of transformations and groups which naturally arise in cryptographic applications. Examples of such transformations and groups are the groups Z2d+×Vn−d+ and Vn−d+×Z2d+$\begin{array}{} \displaystyle \mathbb{Z}_{{2^d}}^{+} \times V_{n - d}^{+} ~\text{and}~ V_{n - d}^{+}\times \mathbb{Z}_{{2^d}}^{+} \end{array}$ and pseudoinversion over the field GF (2 n ) or over the Galois ring GR (2 md , 2 m ).

The independent set problem for a given simple graph is to determine the size of a maximal set of its pairwise non-adjacent vertices. We propose a new way of graph reduction leading to a new proof of the NP-completeness of the independent set problem in the class of planar graphs and to the proof of NP-completeness of this problem in the class of planar graphs having only triangular internal facets of maximal vertex degree 18.

The paper describes structural characteristics of the digraph of an arbitrary polynomial transformation of a finite commutative ring with unity. A classification of vertices of the digraph is proposed: cyclic elements, initial elements, and branch points are described. Quantitative results on such objects and heights of vertices are given. Besides, polynomial transformations are shown to have cycles whose lengths coincide with the lengths of cycles of the induced polynomial transformation over the field R /ℜ, where ℜ is the radical of the finite commutative local ring R .