Volume 31, Issue 5

We consider a fault source under which the fault functions are obtained from the original function f ( x̃ n ) ∈ P2n $\begin{array}{} \displaystyle P_2^n \end{array}$ by a left shift of values of the Boolean variables by at most n . For the vacant positions of the variables, the values are selected from a given filling tuple γ̃ = ( γ 1 , γ 2 , …, γ n ) ∈ E2n $\begin{array}{} \displaystyle E^n_2 \end{array}$, which also moves to the left by the number of positions corresponding to a specific fault function. The problem of diagnostic of faults of this kind is considered. We show that the Shannon function Lγ ~ shifts,diagn(n), $\begin{array}{} \displaystyle L_{\tilde{\gamma}}^{\rm shifts, diagn}(n), \end{array}$ which is equal to the smallest necessary test length for diagnostic of any n -place Boolean function with respect to a described fault source, satisfies the inequality n2≤ Lγ ~ shifts,diagn(n)≤ n. $\begin{array}{} \displaystyle \left\lceil \frac{n}{2} \right\rceil \leq L_{\tilde{\gamma}}^{\rm shifts, diagn}(n) \leq n. \end{array}$

Given a binomial probability distribution on the n -dimensional Boolean cube, the complexity of implementation of Boolean functions by straight line programs with conditional stop is considered. The order, as n → ∞, of the average-case complexity of almost all n -place Boolean functions is established.

The aim of the paper is to find the maximal possible number ξ of units in Boolean triangular array T s formed by s(s+1)2 $\begin{array}{} \displaystyle \frac{s(s+1)}{2} \end{array}$ elements of the field GF (2) defined by the top row of s elements. Each element of each row except the top one is the sum (as in the Pascal’s triangle) of two elements of the above row. It is proved that ξ ⩽ ⌈ s(s+1)3 $\begin{array}{} \displaystyle \frac{s(s+1)}{3} \end{array}$⌉ and this value is attained only on triangles having the upper row as the Fibonacci series mod 2.

The limited deficit method is described, which allows constructing new orthomorphisms (almost orthomorphisms) of groups with the use of those already known. A class of transformations is described under which the set of all orthomorphisms (almost orthomorphisms) remains invariant. It is conjectured that the set of all orthomorphisms (almost orthomorphisms) is generated by transformations implemented by the limited deficit method. This conjecture is verified for all Abelian groups of order at most 12. The spectral-linear method and the spectral-differential method of design of permutations over the additive group of the field 𝔽 2 m ( m = 4, …, 8) are used to construct orthomorphisms with sufficiently high values of the most important cryptographic parameters.

The complexity of implementation of a threshold symmetric n -place Boolean function with threshold k = O (1) via circuits over the basis {∨, ∧} is shown not to exceed 2 log 2 k ⋅ n + o ( n ). Moreover, the complexity of a threshold-2 function is proved to be 2 n + Θ ( n $\begin{array}{} \sqrt n \end{array} $), and the complexity of a threshold-3 function is shown to be 3 n + O (log n ), the corresponding lower bounds are put forward.

We study properties of a p -type subcritical branching process in random environment initiated at moment zero by a vector z = ( z 1 , .., z p ) of particles of different types. For p = 1 the class of processes we consider corresponds to the so-called strongly subcritical case. It is shown that the survival probability of this process up to moment n behaves as C ( z ) λ n for large n , where the parameters λ ∈ (0, 1) and C ( z ) ∈ (0, ∞) are explicitly described in terms of the characteristics of the process. We also demonstrate that the distribution of the number of particles of different types at moment n → ∞ (given its survival up to this moment) does not asymptotically depend on the number and types of particles initiated the process.