Volume 24, Issue 5

The paper is concerned with subcritical branching process in random environment. It is assumed that the moment-generating function of steps of the associated random walk is equal to 1 for some positive value of the argument. Functional limit theorems for sizes of various generations and passage times to various levels are put forward.

McEliece cryptosystem is an public-key cryptosystem; its security is based on the complexity of decoding problem for an arbitrary error-correcting code. V. M. Sidel’nikov in 1994 suggested to construct the cryptosystem on the base of binary Reed-Muller code. In 2007 L. Minder and A. Shokrollahi had designed a structural attack on theMcEliece cryptosystem based on the Reed-Muller codes.Herewe improve their attack and suggest a polynomial attack on the McEliece cryptosystem based on Reed-Muller codes RM(r, m) such that GCD(r,m − 1) = 1.

The hash algorithms of the MDx family involve cyclic shifts, computation of primitive Boolean functions, and addition of constants. So far, very few works have been published in which the authors attempt to explain the impact that the choice of constants, shifts, and Boolean functions has on the cryptographic properties of the algorithms. G. A. Karpunin and H. T. Nguyen suggested a model in which the resistance against differential cryptanalysis may be quantitatively estimated in terms of the number of solutions of a special equation. In this work, in the framework of the aforementioned model, an equation for the MD5 hash function is derived. Examination of one Boolean function and one value of the cyclic shift through exhaustive search requires 2 128 operations of computation of the step of the hash function. The formulas suggested in the present work allow to reduce the complexity of the examination to 2 44 arithmetic operations.

Cooperative games on undirected graphs are considered. The characteristic function is defined in a specialway with due account of the number of links and distances between vertices (players).Animputation procedure is put forward for a tree graph, the resulting imputation is shown to agree with theMyerson vector. For its calculation the use of the generating function is proposed. It is shown that the results obtained may be extended to games with arbitrary graphs.

The paper is concerned with the numbers of non-occurring values in intervals of a stationary sequence of m-dependent random variables with N outcomes. These numbers are shown to be asymptotically joint normally distributed, provided that the number N and the lengths of intervals tend to infinity with the same asymptotic order of growth.