The objective need to improve information security systems in the conditions of information-telecommunication technologies developing is shown. A new area of NP-complete promplems of diophantine analisys is firstly researched, namely Tarry–Escott multigrade diophantine equation systems (MDES) of given dimension and degree. The theorems, that describe parametrical MDES parametric solution properties, required for the development of the algorythm of building of dissymmetric bigram cryptosystem, are given in the article. The author's theorem, which allows to develop information protection system (IPS) mathematical model as a cortege, containing diophantine probleme. A new approach to develop dissymmetric bigram cryptosystem(DBC) on the base of two-parametrical MDES solution, which generalize open key cryptosystem, is considered. Forward and reverse conversions based on parametrical solution are suggested tyo realize on such algorythm. A new concept of equivalence of ordered sets of numbers (parameters) of a given dimension and order is introduced to build DBC mathematical model, using two-parametrical MDES solution. Described mathematical model demonstrates application potential of diophantine equations for IPS with a high degree of reliability.

Keywords: NP-complete probleme, information protection system, symmetric (dissymmetric) cryptosystem, open key cryptosystem, forward and reverse information conversions, Tarry–Escott multigrade diophantine equation system, parametrical solution

The problem of searching over encrypted data using homomorphic encryption in distributed software systems is addressed. The paper considers the application of the approach developed by the authors for the implementation of a distributed system prototypes for issuing digital diplomas and certificates for obtaining competencies with recording in IPFS and an Ethereum smart contract and a search engine for recruiting personnel. The prototype of the system presented in the article demonstrates the use of homomorphic encryption, which allows the search engine and users to interact with the directly transformed data in public blockchain networks without performing the reverse transformation, ensuring their security.

Keywords: blockchain, distributed software systems, cloud computing, homomorphic encryption, mathematical model, transactions, smart contracts, digital diploma, recruiting

The paper demonstrates the objective need of improvment of information security systems in terms of development of information and telecommunications equipment. The theorems that describe the properties of parametric solutions of multigrade diophantine equation systems (MDES) is presented. The properties are used to develop information security system mathematical models. Also the paper demonstrate generalized Frolov's theorem and author's theorem to enable development information security system mathematical models with Diophantine problems. Author's alphabetical information security system mathematical model in form of tuple is given. Approch to develop disymmetric bigramm information security system (DBISS) based on two-parametric solutions is given. The approach generalize establishing open-key information security system. The direct and inverse transformations is realized through parametric solution that provisionally divided into two parts - for direct transformation and inverse transformation. Also author introduces new concept of equity of number ordered set or set of parameter with given dimension and grade. The DBISS mathematical model is given in the paper. The model is based on the MDES two-parametric solution with given dimension and grade. The decribed mathematical model demonstrate the potentil of diophantine equations to development of information security systems with high reliability.

Keywords: information security system, symmetric cryptosystem, disymmetric cryptosystem, open-key cryptosystem, ecncryption, decryption, multigrade diophantine equation system, parametirc solution, diophantine problems