"Exotic" binary number systems for rings of Gauss and Eisenstein integers

Abstract

The paper considers nonstandard binary number systems for rings of Gauss and Eisenstein integers. The principal difference ("exoticism") of such number systems from the canonical number systems introduced by I. Katai for quadratic fields is that as a binary "digital alphabet", it uses a two-element set that does not contain a numeric zero. The paper also synthesizes algorithms for the representation of numbers in the considered number system and characterizes the possibility of an efficient implementation of arithmetic operations.