A mathematical framework for algorithm-based fault-tolerant computing over a ring of integers

Abstract

In this work, we establish a complete mathematical framework for algorithm-based fault-tolerant computing for data vectors defined over a ring of integers. The ring of integers consists of integers {0,1,…,M−1} and all the arithmetic operations are performed modulo the integerM, which is assumed to be composite. The importance of the work lies in the suitability of modulo arithmetic in certain computational environments. Lack of an underlying Galois field,GF(q), presents a unique challenge to this framework. We develop the theory and algorithms for single as well as multiple fault correction and detection. We also analyze the parallel and serial nature of the encoder and decoder configurations. Certain known but rather old results in the theory of numbers dealing with linear congruences and matrix algebra are also described and extended further using mathematical terminology that modern-day researchers are expected to be familiar with.