Volume 16, Issue 5

Let p ≥ 3 be a prime number. In this paper we construct two families of nonlinear p -ary codes derived from the corresponding families of modified Butson–Hadamard matrices. These codes have the minimal distances close to the Plotkin bound and very easy construction and decoding procedures. Moreover, for some of these codes the Plotkin bound is attained.

The algebraic method is widely used in analysis of filter generators of pseudo-random sequences. It is based on obtaining low-degree Boolean equations in bits of the initial states of the generator. The problem to obtain such equations reduces to finding low-degree annihilators for the filtering Boolean function. The presence of nonzero low-degree annihilators decreases the complexity of determining the initial state of the generator by means of its output. In this research we deal with the problem to find all low-degree annihilators for a Boolean function defined as a polynomial in several variables. We propose two new algorithms which solve this problem. Their complexities are bounded above by polynomials of the number of variables of the function and of the number of monomials in the polynomial which defines the function. We also consider the application of these algorithms to realising the algebraic method by three known scenarios which yield low-degree equations.

Boolean functions have found a widespread use in cryptography. In connection with the advent of the 'algebraic' attack on stream ciphers, the Boolean functions, which are used in these ciphers as nonlinear filters, have to possess, among other properties, high algebraic immunity. One more cryptographically significant property of Boolean functions, especially of those utilised in stream ciphers, is nonlinearity. In this connection, the question arises about relations between the nonlinearity of a Boolean function and its algebraic immunity. In this research we obtain a lower bound for nonlinearity in terms of algebraic immunity and present functions at which this bound is attained for any admissible values of the parameters.

We define the distance d ( z , z ′) between points z = x + iy and z ′ = x ′ + iy ′ in the upper half-plane, setting where u = | z – z ′| 2 /( yy ′). The circle K ( z 0 , T ) with centre in a point z 0 consists of the points z satisfying the inequality d ( z , z 0 ) ≤ T . Let N ( z 0 , T ) be the number of elements γ of the modular group PSL 2 ( Z ) such that the point yz 0 lies in the circle K ( z 0 , T ). In this paper, we refine the remainder term in the asymptotic formula for N ( z 0 , T ).

We suggest a new approach to the study of the structural properties of random permutations based on defining some general parametric measure on the set S n of all permutations of degree n which assigns to a permutation with cyclic structure a = ( a 1 , . . . , a n ) the probability proportional to , where θ = ( θ 1 , . . . , θ n ) is a parameter of the measure.

We consider the random variable equal to the number of trees of given size in the graph of a random one-to-one mapping of an n -element set into itself with m connected components. We obtain limit theorems which describe the distribution of this characteristic in the case where n → ∞, m / ln n → ∞, m / ln n = O (ln n ).

We introduce a class of cell circuits, T -circuits, and describe a connection between the lower bound for the area and the depth of the circuits of this class: the less the depth the greater the area of a circuit. We give examples of T -circuits with logarithmic depth in the problem of calculation of n prefix sums and also of sums and differences of two n -digit numbers. It is shown that the area of these circuits is O ( n log n ) and has the optimal order.

We give an example of a quasi-universal function in the class ε 2 of the Grzegorczyk hierarchy. This function is of very simple structure and does not contain an explicit enumeration of any Turing machine. As a corollary we obtain a simple basis over superposition in the class ε 2 .