Algorithms for determining residues modulo in a complex numerical domain

An important aspect of improving modern computer systems and their components is an increasing the speed of arithmetic calculations, including due to the use of new mathematical models and methods based on non-positional residue number systems. The increase in the volume of processed data in modern computer systems leads to the additional risks and threats of unintentional failures and denials of service. This is especially important when building fault-tolerant critical information systems in which failure or denial of service can lead to catastrophic consequences. The article discusses arithmetic operations in the ring of residue classes. These techniques make it possible to implement fast and fault-tolerant computing for modern computer systems and telecommunication networks. We propose an algorithm for calculating the residues of integer data in a complex numerical domain. The algorithm is based on the use of the first fundamental Gauss theorem, which establishes an isomorphism between complex and real residues. Examples of determining the residues of integer data in a complex numerical domain are presented, which clearly demonstrate the constructiveness of the proposed techniques.