ARITHMETIC OF ASYMMETRIC CRYPTOSYSTEMS IN THE FIELD OF COMPLEX NUMBERS

Abstract

At the current stage of information technology development, there is a need to improve existing and develop new methods and means of increasing the productivity of asymmetric crypto-algorithms. The article develops the theoretical foundations of modular calculations and asymmetric cryptography in the complex numerical domain. The method of determining the complex and real residues based on the complex module is considered. Euclid's algorithm and its consequence for finding an inverse element in a complex numerical domain are considered. A comparison of the complexity of Euclid's algorithm for finding the inverse of the element when finding the smallest positive and absolutely smallest residues was made. An analogue of Euler's function in the complex numerical domain was searched and this function was used to find the inverse of a complex number. The restoration of a complex number using the Chinese remainder theorem is demonstrated. The considered modular calculations in the field of complex numbers can be used in the construction of new approaches to asymmetric encryption.