Method for performing the operation of adding the remainder of numbers modulo

Abstract

One of the components of a computer system (CS) in a positional binary number system (PNS) is an adder of two numbers. In particular, adders modulo m_i of two numbers are also components of the CS. This type of modulo adders is widely used both in the PNS and in the non-positional number system in the residual classes (RNS). An  important and urgent scientific and applied problem is the problem of constructing the adders, which operate by modulus m_i, that is an arbitrary RNS modulo. If the remainders a_i and b_i  of both numbers  A and B  in RNS are represented in a binary PNS, then the adder of two residuals a_i and b_i by modulus m_i is a sequential set of n binary one-bit adders (BOBA).  The purpose of the article is to develop a method for performing the operation of modular addition (a_i + b_i)mod m_i of two remainders of numbers by an arbitrary modulo m_i based on the use of a positional binary adder modulo M = 2ⁿ - 1. The proposed method is based on the use of the well-known structure of positional binary adders modulo M = 2ⁿ - 1. Technically, the problem of creating the structure of the modular adder is formulated as follows. It is necessary to provide conditions under which the initial adder in PNS modulo M  performs the addition operation modulo m_i.  This procedure is carried out by introducing additional connections as X_¯i­j in the positional adder modulo M = 2ⁿ - 1, where the expression X_¯i­j denotes one-way connection between the output of the j-th BOBA and the input of the i-th BOBA.