Mathematical model of information security systems based on Diophantine sets

Development of the asymmetric cryptography started with the appearance of the first knapsack information protection system, when, in 1978, Ralph Merkel and Martin Hellman proposed to use different keys for forward and reverse mapping data for encryption. Now this model, like many based on are considered to be insecure. As a result the authority of knapsack systems was low. However, some of these systems are still considered persistent, for example, the model proposed in 1988 by Ben Shore and Ronald Rivest.
In the article stated and solved the problem of argumentation of cryptographic strength of the non-standard knapsack information security systems. Justified diophantine difficulties that arise in the study of vulnerabilities of the investigated information security systems. Revealed the qualitative features of non-standard knapsack systems that increase their resistance to known attacks. In this paper, we propose a mathematical model of polyalphabetic cryptosystem, in which the algorithm of inverse transformation of closed text is algorithmically unsolvable problem for the analyst. It's permeated with the idea K. Shennon, who believed that cryptosystems, containing Diophantine problems, have the greatest variation in the selection of key.