Volume 35, Issue 4

Finite quasigroups and n -quasigroups are a promising platform for cryptoalgorithm implementation. One of the key problems consists in memory-efficient generation of wide classes of n -quasigroups of a large order. We describe a possible solution based on proper families of functions, show that the number of n -quasigroups generated thereby is bounded from below in terms of the cardinality of the image of the corresponding proper family, study possible values that this cardinality can take, and give two examples of quadratic proper families of Boolean functions with a high image cardinality.

A class of Boolean functions constructed from digital sequences of linear recurrences over the ring Z2n $\mathbb{Z}_{2^n}$ is considered. We investigate distances between functions, the cardinality of the class, nonlinearity and weights of functions. It is shown that this class consists of functions that are rather distant from the class of all affine functions.

In this paper we investigate the propagation criterion for strict majority symmetric Boolean functions. With the use of the theory of Krawtchouk polynomials it is shown that vectors whose Hamming weight differs from n /2 by at most 1/2 satisfy the propagation criterion for strict majority functions in n variables, where ⌊ n /2⌋ is odd.

It is well known that a branching process in random environment (BPRE) can be analyzed via the associated random walk Sn=ξ1+⋯+ξn, $$\begin{equation*}S_n = \xi_1 + \dotsb + \xi_n,\end{equation*}$$ where ξk=lnφηk′ $\xi_k = \ln \varphi_{\eta_k}'$(1). Here {ηk}k=1∞ $\{ \eta_k \}_{k = 1}^{\infty}$ is the random environment and φ x ( t ) is the generating function of the number of descendants of a particle for given environment x . We study the probability of extinction of a branching process in random environment with freezing: in constrast to classic BPRE, in this process every state η k of the environment lasts for given number τ k of generations. It turns out that this variant of BPRE is also closely related to a random walk Sn=τ1ξ1+⋯+τnξn. $$\begin{equation*}S_n = \tau_1 \xi_1 + \dotsb + \tau_n \xi_n.\end{equation*}$$ We find several sufficient conditions for extinction probability of such process to be one or less than one correspondingly.

In the case when the tested sequence consists of independent random variables having a Bernoulli distribution with the parameter p=12 $p=\frac{1}{2}$ the limit joint distribution of the statistics T 1 , T 2 , T 3 of the following three tests of the NIST package is obtained: «Monobit Test», «Frequency Test within a Block» and «Binary Matrix Rank Test». Necessary and sufficient conditions for asymptotic uncorrelatedness and/or asymptotic independence of these statistics are obtained. It is proved that the covariance matrix C = ‖ C ij ‖ of the limit distribution of the vector ( T 1 , T 2 , T 3 ) satisfies the relations C 12 = C 21 = C 13 = C 31 = 0, C 23 = C 32 ≥ 0. The limit behavior of the vector ( T 1 , T 2 , T 3 ) is described for a wide class of values p≠12. $p\neq\frac{1}{2} .$

Let 𝔖 n be a semigroup of mappings of the n -element set X into itself and 𝔖 n ( A ) be a subset of mappings from 𝔖 n such that sizes of all their components belong to the set A . By σ n = σ n ( A ) we denote a random mapping having uniform distribution on the set 𝔖 n ( A ). Such mappings were considered by A. N. Timashev in 2019. For some class of sets A having positive densities in the set N of natural numbers the asymptotic of cardinalities of sets 𝔖 n ( A ) is found for n → ∞. Also the estimate for the total variation distance between the structure of the mapping σ n ( A ) and corresponding sequence of independent Poisson random variables is obtained.