Volume 33, Issue 1

We identify the classes of Boolean functions that may be implemented by easily testable circuits in the Zhegalkin basis for constant type-1 faults on outputs of gates. An upper estimate for the length of a complete fault detection test for three-place functions is obtained.

The asymptotic behavior of general independence numbers of random hypergraphs for the binomial model is studied. We prove that for some types of parameter variations the distribution of independence numbers is concentrated on two neighboring values.

We show that the system of elementary conjunctions Ωn,2k=K0,…,K2k−1 $ \Omega_{n,2^k} = {K_0,\ldots,K_{2^{k} -1}} $ such that each conjunction depends essentially on n variables and corresponds to some codeword of a linear ( n , k )-code can be implemented by a separating contact circuit of complexity at most 2 k +1 +4 k ( n  −  k ) − 2. We also show that if a contact (1, 2 k )-terminal network is separating and implements the system of elementary conjunctions Ωn,2k $ \Omega_{n,2^k} $ , then the number of contacts in it is at least 2 k +1 − 2.

A classical scheme of random equiprobable allocations of n particles into N cells is considered. We find an asymptotic formula for the probability that the number of empty cells is equal to k under the condition that n , N  → ∞ in such a way that n /( N  −  k ) is bounded and separated from 1 from below.

An n -place function over a field Fq $ \mathbf {F}_q $ with q elements is called maximally nonlinear if it has the largest nonlinearity among all q -valued n -place functions. We show that, for even n =2, a function is maximally nonlinear if and only if its nonlinearity is qn−1(q−1)−qn2−1 $ q^{n-1}(q - 1) - q^{\frac n2-1} $ ; for n =1, the corresponding criterion for maximal nonlinearity is q  − 2. For q>2 $ q \gt 2 $ and even n =2, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions.

For sequences of independent random variables having a Bernoulli distribution with parameter p the limit joint distribution of statistics of four tests of the NIST statistical package (« Monobit Test », « Frequency Test within a Block », « Runs Test » and a generalization of « Non-overlapping Template Matching Test ») is obtained. Conditions of asymptotic uncorrelatedness and/or asymptotic independence of these statistics are given.