Volume 29, Issue 2

We describe an algorithm extending the previously proposed method of key recovery of a filter generator. The algorithm is based on an approximation of the combining function by algebraically degenerate functions. We give estimates of the computational complexity, reliability, and the amount of memory used by the method. Examples of application of the method are considered, in particular, for the analysis of the LILI-128 cipher.

A new approach to the study of algebraic, combinatorial, and cryptographic properties of Boolean functions is proposed. New relations between functions have been revealed by consideration of an injective mapping of the set of Boolean functions onto the sphere in a Euclidean space. Moreover, under this mapping some classes of functions have extremely regular localizations on the sphere. We introduce the concept of curvature of a Boolean function, which characterizes its proximity (in some sense) to maximally nonlinear functions.

The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule A M B over left- and right-Artinian rings A and B , respectively, is associated with the equivalent bimodule of translations C M Z , where C is the multiplicative ring of the bimodule A M B and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules A M B and C M Z is studied. It is demonstrated that, in the general case, the fact that A M B is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule C M Z . However, under some additional assumptions it can be shown that if C M Z is a quasi-Frobenius bimodule, then the bimodule A M B is quasi-Frobenius as well.

This article is related to the average estimates of numerical functions Ω ( k ; n ) and ω ( k ; n ) connected with the number of prime divisors of n with limited multiplicity.

The functional system of Boolean vector functions with naturally defined superposition operation is studied. Sufficient conditions for membership of a number of important vector functions in closed classes are given.

For a ring A , the following conditions are equivalent. A is a right strongly semiprime ring. Every right A -module which is injective with respect to some essential right ideal of the ring A , is an injective module. Every quasi-injective right A -module which is injective with respect to some essential right ideal of the ring A is an injective module.