Synthesis of combinational circuits using Galois field arithmetic

Abstract

A Fourier-like transform for combinational switching functions of n-input m-output (n≥m) is presented.The transform utilizes Galois fields of 2n and 2m elements.Operations between these fields are defined suitably to retain generality. Application of this transform to swithing function results in a polynomial expansion which leads to a direct realization of the function.Logic circuits so realized are composed of complex modules that perform arithmetic operations in Galois fields and are suitable for fabrication by LSI technology.In general, such logic circuits assume 2n different levels.The complexity of the circuits grows faster than n2.2n.This high degree of complexity is due to rather inefficient utilization of modules.The complexity of the circuits so synthesized is reduced by repeated use of the modules.Such use is facilitated by an arithmetic-unit like structure with some dynamic storage capability.Coefficients of various terms in the expansion are stored in the storage.These coefficients are made available at proper time in a sequential manner during generation of the polynomial.The delay through such circuits is inherently large.Considerable reduction in the delay can be achieved by adding little hardware.External capability to change contents of the storage makes such circuits universal in nature. Universal logic blocks so constructed require 2n+3 pins to realize any multi-output function of n variables (n≥m).Application and development of the transform require computation in Galois fields.A brief discussion of mathematical nature and properties of such fields is included to make a self-contained presentation.