Volume 19, Issue 6

In this paper, we develop the fingerprinting technique of calculation of Boolean functions in quantum calculation models. The use of the fingerprinting technique is demonstrated on the example of calculation of the function MOD m in the class of quantum OBDD (oblivious read-once branching programs). Next, the potentialities of the fingerprinting technique are demonstrated in the example of realisation of the ‘equality’ function in the quantum model of communication with a referee (the SMP communication model) and in examples of recognition of languages in quantum automata. All given realisations of Boolean functions in various quantum calculation models (OBDD, SMP communication model, and a finite automaton) are asymptotically optimal. The authors thank Juhani Karhumäki for the invitation to the University of Turku and fruitful discussions of this research.

We calculate the exact values of the probabilities of the active attacks for an authentication code with two states of a source.

We analyse the model of embedding of secret information into a Bernoulli sequence transmitted, where each element of the initial sequence is replaced with some probability by a certain Bernoulli symbol. We find the conditions for the parameters of the model which guarantee the secrecy of the fact of embedding. A similar analysis is carried out for the polynomial model.

We find formulas for the number of solutions of equations in finite groups and with the use of these results define naturally the tpp -groups, which play an important role in the theory of fast multiplication of matrices.

The zero-divisor graph of an associative ring R is a graph whose vertices are all nonzero (one-sided and two-sided) zero divisors of R , two distinct vertices x, y are connected by an edge if and only if xy = 0 or yx = 0. In this paper, all finite nonnilpotent rings with planar zero-divisor graphs are completely described. In the previous paper by Kuzmina and Maltsev, the finite nilpotent rings with planar zero-divisor graphs were studied. Thus, this paper completes the description of finite rings with planar zero-divisor graphs.

All finitely generated right A -modules are ℵ 0 -injective if and only if A is a regular, right ℵ 0 -injective ring.

The notion of a boundary class is a useful notion in the investigation of the complexity of extremal problems on graphs. One boundary class is known for the independent set problem and three boundary classes are known for the dominating set problem. In this paper it is proved that the set of boundary classes for the 3-colouring problem is infinite.

In this research we give lower bounds for periods of output sequences of automata for a given periodic input sequence.

We consider generalised polarised polynomials for k -valued functions (for prime k ). It is proved that each k -valued function is represented by some unique generalised polarised polynomial for each polarisation vector. We find upper and lower bounds for the Shannon functions of degree and length of the generalised polarised polynomials of k -valued functions.