Volume 23, Issue 1

For the complexity of multiplication in a standard basis of the field GF(2 n ), where n = 2·3 k , the upper bound 5n log 3 n log 2 log 3 n + O(n log n) for multiplication complexity and an asymptotically 2.5 times greater bound for inversion complexity are obtained. As a consequence, for the complexity of multiplication of binary polynomials the upper bound (10 + o (1)) N log 3 N log 2 log N is valid. The mentioned estimates are generalised to some other finite fields. Let n = (p - 1)p k , where p is a prime such that 2 is a primitive root modulo p, 2 p-1 - 1 is not divided by p2. For a standard basis of the field GF(2 n ), we construct a multiplier of complexity (C+o(1))( n log n log log n ) and an inverter of complexity O (log p) n log n log log n , where C≤10. This research was supported by the Russian Foundation for Basic Research, grants 11-01-00508 and 11-01-00792-a, and by the Program of Fundamental Research of the Department of Mathematical Sciences of the Russian Academy of Sciences ‘Algebraic and Combinatorial Methods of Mathematical Cybernetics and Information Systems of New Generation,’ project ‘Problems of Optimal Synthesis of Control Systems.’

In this research we consider realisations of linear Boolean functions by circuits of functional elements over the basis B which consists of two-input functional elements realising nonlinear functions and an inverter. The weight of a two-input element over the basis B is set equal to one, and the weight of the inverter is zero. We give a description of all minimal circuits realising linear function over the basis B and show that all these circuits are of specific block form. This research was supported by the Russian Foundation for Basic Research, grant 11- 01-00508, and the Program of Basic Research of the Department of Mathematical Sciences of the Russian Academy of Sciences ‘Algebraic and Combinatorial Methods of Mathematical Cybernetics and Information Systems of New Generation,’ project ‘Problems of Optimal Synthesis of Control Systems.’

We consider the characteristic n ( k ) max (m), which is the maximum number of variables in a Boolean function of fixed weight m possessing no implicent of k variables. The obtained results on the structure of the set of satisfying vectors of such functions allow us to estimate and, in some cases, find exact values of n ( k ) max (m) for m < 3·2 k-1 , k ≥ 3.

Problems of k -valued logic generalisations of the Goli´c conjecture are considered. In the case of prime k , a number of results on correctness of the k -valued analogue of the Goli´c conjecture are obtained in many subcases. We also present a full proof of incorrectness of the k -valued analogue of the Goli´c conjecture in the case of composite k . This research was supported by the Russian Foundation for Basic Research, grant 12- 01-00680-a.

We analyse properties of the Walsh spectrum of Boolean functions of n variables which are chosen at random from some subsets of the set of all these functions. We derive the characteristic functions of the spectrum and find exact and asymptotic, as n → ∞, distributions of its various characteristics.

We study modules M such that all automorphisms of submodules in M can be extended to endomorphisms of M. This research was supported by the Russian Foundation for Basic Research, grant 08- 01-00693-a.