A Method for Arithmetic Comparison of Data Represented in a Residue Number System
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NEW MEANS OF CYBERNETICS, INFORMATICS, COMPUTER ENGINEERING, AND SYSTEMS ANALYSIS A METHOD FOR ARITHMETIC COMPARISON OF DATA REPRESENTED IN A RESIDUE NUMBER SYSTEM V. A. Krasnobayev,1 A. S. Yanko,2 and S. A. Koshman3
UDC 681.04
Abstract. This paper considers methods for arithmetic comparison of data represented in a residue number system (RNS). A mathematical model and a method for exact arithmetic comparison of data in RNS are developed. The proposed method is based on the obtainment and use of position signs of position-independent codes and maximizes the validity of comparing numbers in RNS with a minimum amount of the equipment implementing the method. Keywords: computer system for processing integer data, residue number system, method for comparing data, accuracy of comparing integers, position sign of position-independent code. INTRODUCTION As is well known, the prime advantage of a position-independent residue number system (RNS) is the possibility to organize the process of fast processing of integers. The use of RNSs makes it possible to create methods and digital hardware of computer systems that improve user efficiency in solving definite classes of problems in which the operations of integer arithmetic addition, subtraction, and multiplication are applied. This is reached owing to the use of RNS properties such as independence, equality, and small length of residues whose totality {a i } represents a number ARNS = ( a1 || a 2 || K || a i -1 || a i || a i + 1 ||K || a n ) in terms of n bases (moduli) of a given position-independent number system [1, 2]. The need for solving a wide class of problems containing logical operations (for example, the operation of comparison of two numbers ARNS = ( a1 || a 2 || K || a i -1 || a i || a i + 1 ||K || a n ) and B RNS = ( b1 || b2 ||K || bi -1 || bi || bi + 1 ||K || bn ) , which often occurs in control problems) along with arithmetic integer operations by a computer system that processes integer data (CSPID) and perform operations on RNS numbers reduces the overall efficiency of using a position-independent number system. This is stipulated by a considerable (in comparison with the execution of the above-mentioned arithmetic modular operations) time of execution of the data comparison operation in RNS. Therefore, the investigation and improvement of the existing methods and algorithms for hardware implementation of the operation of arithmetic comparison of data in RNS and also the development of new ones is an important and topical scientific and applied problem of creation of CSPIDs. METHODS FOR COMPARING NUMBERS IN A RESIDUE NUMBER SYSTEM As is well known, there are three groups of methods for comparing numbers in RNS [3, 4]. The methods of direct comparison belong to the first group. They are based on the transformation of numbers ARNS and  RNS from an RNS 1
V. N. Karazin Kharkiv National University, Kharkiv, Ukraine, [email protected]. 2Poltava National Technical Yuriy Kondratyuk University, Poltava, Ukraine, [email protected]. 3Kharkiv Petro Vasilen
