Разработка математических моделей систем защиты информации на основе многостепенных систем диофантовых уравнений

Abstract

The objective necessity of improvement of information security systems (SPI) in the conditions of development of information and telecommunication technologies is shown. Theorems which allow to describe the properties of parametric solutions of multistage systems of Diophantine equations necessary for the development of mathematical models of SPI on their basis are given. The theorem generalizing the known Frolov's theorem is presented, and the author's theorem on the basis of which the mathematical model of SPI containing Diophantine difficulties is developed is given. A new approach to the development of SPI generalizing the principle of construction of public key cryptosystems is proposed. One part of the conditional identity is used for the direct transformation of the original message, and the other part - for the inverse transformation. A new concept of equivalence of ordered sets of numbers or parameters with a given dimension and degree is introduced. Mathematical models of cryptosystems developed on the basis of two-parameter solutions of multistage systems of Diophantine equations, in particular, - equations of the fifth degree with the number of variables equal to twelve are presented. The described mathematical models demonstrate the potential of using Diophantine equations for the development of SPI with a high degree of reliability. These models allow to build asymmetric GIS, and the system public key. Such systems contain Diophantine difficulties admitting the existence of a countable set of equally probable keys.