Band 19, Heft 5

The McEliece cryptosystem is one of the oldest public-key cryptosystems. It was proposed by R. J. McEliece in 1978. The McEliece cryptosystem is based on a certain NP -hard problem in the coding theory. In this paper we consider a generalisation of the McEliece cryptosystem proposed by V. M. Sidelnikov in 1994. The McEliece–Sidelnikov cryptosystem is based on u -fold use of the Reed–Muller codes RM ( r, m ). This research is devoted to questions related to the space of equivalent secret keys of the new cryptosystem, that is, secret keys generating identical public keys. We study the structure of the set of public keys of the McEliece–Sidelnikov cryptosystem for an arbitrary number u of blocks. For the case of two blocks we describe all equivalence classes of secret keys with representatives of a special form.

We develop the game-theoretical approach to estimation of strength of authentication codes suggested by G. J. Simmons. It is proved that a combination of imitation and substitution attacks allows the opponent to get a greater gain than that from separate imitation or substitution attacks.

We obtain a representation of the number K n of repetition-free Boolean functions of n variables over the elementary basis {&, ∨,¯} in the form of a convergent exponential power series. This representation is the simplest representation among a number of similar formulas containing different combinatorial numbers. The obtained result gives a possibility to find the asymptotics of K n .

One of the key results of the theory of matroids is the well-known Rado–Edmonds theorem which states that the problem of maximising an additive function on an independence system is solvable by the greedy algorithm if and only if the independence system is a matroid. Many generalisations of the Rado–Edmonds theorem were related in main with extension of the set of feasible solutions of the maximisation problem, as a rule an independence system was replaced by an accessible set system. Extension of the class of objective functions was also accompanied by extension of the feasible sets. In this paper, we investigate the maximisation problems of nonadditive functions of sets on independence systems. We suggest a characterisation of the class of objective functions of the problems on matroids which are solvable by the greedy algorithm. We proved a generalisation of the Rado–Edmonds theorem for the maximisation problem on the independence system of a nonadditive function of a given class.

We consider a table with columns each of which contains each symbol of an alphabet A precisely once; the remaining elements of a column are equal to zero. It is required to transform the table preserving the sets of elements in each row and each column to the form with consecutively placed elements of alphabet A in each row. We give conditions for solvability of this problem.

Let c ≥ 2 and p ≥ c be two integers. We say that a proper colouring of the graph G is ( c, p )-nondegenerate, if for any vertex of G of degree at least p there are at least c vertices of different colours adjacent to it. In this research we prove the following result which generalises the Brooks theorem. Let D ≥ 3 and G be a graph without cliques on D + 1 vertices and a degree of any vertex in this graph be no greater than D . Then for any integer c ≥ 2 there is a proper ( c, p )-nondegenerate vertex D -colouring of G , where p = ( c 3 + 8 c 2 + 19 c + 6)( c – 1).