Hybrid Number Systems: Application for Calculations in Galois Fields

Abstract

Hybrid-base numbering systems are proposed; each digit of a number recorded in the system corresponds to some prime number. Subdivisions using for recording of a number in the frames of proposed approach correspond to specific Galois fields that correspond to the prime numbers that form the base of the coding system. Division of a digit into subunits is based on the theory of algebraic rings, which allows the number to be represented as a linear combination of idempotent elements with coefficients corresponding to the elements of the Galois fields. A specific example of binary-ternary logic where operations are performed in a number system with a base equal to six is considered. It is shown that operations in the decimal number system can similarly be reduced to operations in binary-ternary logic. The issues of practical use of algorithms based on hybrid number systems are discussed. In particular, a scheme of an adder operating with binary-ternary logic is proposed. The advantages of the proposed approach in terms of the possibility of its use in computing machinery on innovative element base are discussed.