Volume 25, Issue 3

The paper is concerned with the process of fluctuation of the player rating in the Elo model in an infinite series of games with the same adversary. Under some assumptions on the model parameters, this process is shown to have a stationary distribution; its median is found for players of the same skill.

Polynomial normal forms for functions of κ-valued logics (for prime values of κ) are considered. A polynomial normal form modulo κ (p.n.f.) is the sum modulo κ of products of variables, or variables with one or several Post negations, taken with some coefficients. The length of a p.n.f. is the number of distinct terms that appear in the form with nonzero coefficients. If κ is a prime number, then each function of κ-valued logic may be represented by various p.n.f. The length of a function of κ-valued logic in the class of p.n.f. is the minimum length of a p.n.f. representing this function. The Shannon function L p.n.f. κ (n) of the length of functions of κ-valued logic in the class of p.n.f. is the maximum length of a function in the class of p.n.f. among all n-ary functions of κ-valued logic. In thiswork the order of growth of the Shannon function L p.n.f κ (n) is established (for prime values of κ):

We investigate a multitype critical branching process in an i.i.d. random environment. A functional limit theorem is proved for the logarithm of the number of particles in the process at moments nt, 0 ≤ t ≤ 1, conditioned on its survival up to moment n → ∞.

An estimate of the accuracy of the Poisson approximation in B. A. Sevastyanov’s theorem providing conditions for the distribution of the sum of random indicators to converge to the Poisson distribution is obtained. This result is applied to estimate the rate of convergence to the limit Poisson distribution in a theorem on the number of solutions of systems of random inclusions.

We investigate a previously proposed attack against the public key cryptosystem MST 3 based on a Suzuki 2-group. We present a theoretical foundation of practical results obtained by the authors of the considered attack, and show that in the general case this attack may have significant complexity. Besides that we investigate the set of logarithmic signatures of the centre of a Suzuki 2-group and estimate the cardinality of this set.

Let N be a set of N elements and F 1 , F 2 ,... be a sequence of random independent equiprobable mappings N → N. For a subset S 0 ⊂ N, |S 0 | = n, we consider a sequence of its images S k = F k (. . . F 2 (F 1 (S 0 )) . . .), k = 1, 2... , and a sequence of their unions Ψ k = S 1 ⋃ ... ⋃ S k , k = 1, 2 ... An approach to the exact computation of distribution of |S k | and |Ψ k | for moderate values of N is described. Two-sided inequalities for M|S k | and M|Ψ k | such that upper bound are asymptotically equivalent to lower ones for N, n, k → ∞, nk = o(N) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.