Abstract We obtain upper and lower estimates for the difference characteristics of permutations of the field F 2 n $\mathbb{F}_{2^{n}}$ whose restrictions to cosets of the group F 2 n × $\mathbb{F}^{\times}_{2^{n}}$ by its subgroup H , | H | = l , l · r = 2 n − 1, are mappings of the form x → c i x , c i ∈ F 2 n × , i = 0 , … , r − 1. $x\to c_{i}x,\, c_{i}\in\mathbb{F}^{\times}_{2^{n}},\, i=0,\dots,r-1.$

Abstract For Boolean functions f of special form, we obtain an upper estimate for the length D ( f ) of a fault detection test if f is implemented by circuits of gates in the Zhegalkin basis with constant type-1 faults at gate outputs. As a corollary, D ( f ) ≤ n k − 1 ( k − 2 ) ! + 1 $D(f) \leq \frac{n^{k-1}}{(k-2)!}+1$ for functions f of n ≥ k variables whose Zhegalkin polynomial is of degree at most k .

Abstract Aho-Corasick automaton is widely used to find occurrences of words from a given set in a text. In our paper we introduce an equivalence relation ∼ R $\stackrel{R}{\sim}$ on states of Aho–Corasick automaton and prove indistinguishability of ∼ R $\stackrel{R}{\sim}$ -equivalent states. We also propose an algorithm for construction of a ∼ R $\stackrel{R}{\sim}$ -minimal automaton whose states are ∼ R $\stackrel{R}{\sim}$ -equivalence classes. Time and space complexity of this algorithm are linear in the number of states of the original Aho–Corasick automaton. Finally we consider cases in which the relations of ∼ R $\stackrel{R}{\sim}$ -equivalence and indistinguishability are identical, and thus the proposed automaton is minimal.

Abstract We prove that a solution of functional equations of generalized transitivity for strongly dependent binary operations can be described similarly to the case of quasigroups by replacing the emerging group structure with a monoid. We also perform a generalization to the case of n -ary strongly dependent operations.

Abstract We prove that the complexity of computation of the threshold symmetric function T n n − 1 $T_n^{n-1}$ by monotone switching networks is Ω ( n log log n ).

Abstract We consider configuration graphs with N vertices. The vertex degrees are independent identically distributed random variables and for any vertex of the graph the distribution of its degree η satisfies the following condition: P { η = k } ∼ d k g ln h k , k → ∞ , $$\mathbf{P}\{\eta=k\} \sim \frac{d}{k^g \ln ^h k}, \quad k \rightarrow \infty,$$ where d > 0, h ⩾ 0, 2 < g < 3. We obtain the limit distributions of the maximal degree of vertices in the configuration graph as N , n → ∞and n / N (3 g −4)/(2 g −2) → ∞under the conditions that the sum of vertex degrees is n .

Abstract It is proved that the generalized Wiener attack on the RSA cryptosystem allows one to find not only small, but also some large secret exponents d , and the fraction of exponents d which are weak against this attack is heuristically estimated as O ( N −1/2 ).

Abstract A class of Boolean functions constructed from digital sequences of linear recurrences over the ring Z 2 n $\mathbb{Z}_{2^n}$ 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.

Abstract In a general scheme of allocation of no more than n particles to N cells we prove limit theorems for the random variable ηn ,N (K) which is the number of particles in a given set of K cells. The main result of the paper is Theorem 1. Limit distribution in this theorem depends on s = lim K N . $s=\lim \frac{K}{N} .$ If 0 < s < 1, then the limit distribution is that of the minimum of independent Gaussian random variables, and if s = 1, then it is the distribution of the absolute value of a Gaussian random variable taken with the minus sign.