Volume 30, Issue 2

An efficient recursive method for synthesis of correlation-immune Boolean functions is proposed. At the first stage, this method uses minimal correlation-immune functions. A classification of 6-variable minimal correlation-immune functions under the Jevons group is put forward. New results on minimal correlation-immune functions are given.

A new equivalence relation on the set of Boolean functions is introduced: functions are declared to be Δ -equivalent if their autocorrelation functions are equal. It turns out that this classification agrees well with the cryptographic properties of Boolean functions: for functions belonging to the same Δ -equivalence class a number of their cryptographic characteristics do coincide. For example, all bent-functions (of a fixed number of variables) make up one class.

We prove that, for n ⩾ 2, any n -place Boolean function may be implemented by a two-pole contact circuit which is irredundant and allows a diagnostic test with length not exceeding n + k ( n − 2) under at most k contact breaks. It is shown that with k = k ( n ) ⩽ 2 n −4 , for almost all n -place Boolean functions, the least possible length of such a test is at most 2 k + 2.

A dynamic system of cities with migrants is considered. The wage function is each city depends on the number of migrants in the city. The system is modeled by an automaton whose state is the vector consisting of the numbers of migrants in the cities. The transition function of the automaton reflects the conditions for transfers of migrants between cities. The system stabilizes if the moves are stopped at some point. We find conditions for stabilization of such system depending on the restrictions on the wage function and the automaton transition function. It is shown that if the functions of wages are strictly decreasing, if their ranges are disjoint, and if the transition function is defined so that a migrant moves to another city if and only if its salary increases, then the system necessarily stabilizes and its final state depends only on the total number of migrants and does not depend on their initial distribution over the cities. However, if the transition function is changed so that a migrant moves also if its salary is preserved, but the total wages in all cities are increased, then a monotonous decrease in the wage functions is sufficient for stabilization of the system.

We study the discrepancy of linear recurring sequences over Galois rings. By means of an estimate of an exponential sum some nontrivial bounds on the discrepancy are derived. It is shown that these bounds are asymptotically not worse than known estimates for maximal period linear recurring sequences over prime fields.

Models of multi-output and scalar recursive Boolean circuits of bounded depth in an arbitrary basis are considered. Methods for lower and upper estimates for the Shannon function for the complexity of circuits of these classes are provided. Based on these methods, an asymptotic formula for the Shannon function is put forward. Moreover, in the above classes of recursive circuits, upper estimates for the complexity of implementation of some functions and systems of functions used in applications are obtained.