Usage of polynomial representation of numbers for approximate homomorphic encryption

\emph {Introduction} In the modern world of computers and networks the idea of expanding of personal computer resources with the help of cloud storages and computation looks more and more lucrative. However, usage of these resources may endanger data being processed. In last twenty years several algorithms of homomorphic encryption were developed allowing solving of this problem among other applications. However such algorithms are usually constructed as public key systems for long term storage and processing of data. In this article two algorithms of homomorphic encryption optimized for single data processing are proposed. \emph {Purpose} The target of research is development of data coding system which allows safe data processing in public clouds. \emph {Results} Two homomorphic coding systems had been developed, first is based on representation of numbers in the form of polynomials, second based on further representation of polynomials in the form of sets of values. Developed systems allow approximate calculations of coded data without decryption allowing processing of real numbers. System has high level of protection and provides high precision of calculations, comparable with standard personal computer calculation precision. Structure of coded data allows parallel computing. Proposed system allows safe data processing in public networks. Question of finding of optimal parameters for the system stands open both for high precision calculation of limited sets of operations and repeatedly good precision for big sets of operations.