ON UNITS IN ONE CLASS OF GROUP RINGS

Abstract

The basis of the vast majority of cryptographic information protection systems are the so-called computationally hard problems. One such problem is to find the discrete logarithm in a suitably chosen finite group. This problem consists in obtaining for two arbitrary elements of the group such a natural number that the first element to the power of this number is equal to the second element. Currently, research on discrete logarithm-based methods in groups with commutative or non-commutative operation is insufficient. The research of the mentioned issues of information protection is also affected by expectations regarding the appearance of powerful quantum computers that will be able to solve hard computational problems in polynomial time, which are beyond the capabilities of modern deterministic computers. Therefore, in particular, they study groups consisting of units of group rings specified by a certain ring with a unit and a group.

The issue of finding units for partial group rings, which can be defined by any finite field and finite cyclic group, is investigated. A program was developed in the Python environment for performing calculations on the elements of such group rings (raising an element to the power of a large natural number, finding out whether an element is a unit or not a unit, finding the number of different powers of an arbitrary element, factoring of polynomials that define a group ring into irreducible factors). With the use of this program, computational data were obtained, which made it possible to formulate assumptions about the exact number of elements in terms of the number of elements of the corresponding field and group. The application of the specified number will allow finding group ring units that would simultaneously have the large multiplicative order. Actually, these are needed when constructing a series of recently proposed asymmetric cryptosystems.