Fully Homomorphic Encryption for Parallel Implementation of Approximate Methods for Solving Differential Equations

A parallel fully homomorphic encryption for rational numbers is developed in this paper. Parallelism of processing is achieved by using methods of modular arithmetic. Encryption is constructed by mapping the field of rational numbers onto a vector space. Two operations, namely addition and multiplication, are defined. Addition and multiplication tables are constructed, which ensures that a vector space is closed under these mathematical operations. We show the implementation of protected recursive computations in rings of the form ZM, M = m(1)m(2) ... m(k). We give a criterion of effective use of encryption for the numerical solution of the Cauchy problem. It is proved that the efficiency of encryption increases with increasing volumes and accuracy of computations.