In this paper we present several sets of mathematical tools for characterizing the solutions of linear Diophantine equations. First, a number of methods are given for reducing the complexity of the computations. Thereafter, we introduce different techniques for determining the exact number of solutions of linear Diophantine equations. Finally, we present a method for extracting efficiently the solutions of such equations. For all these methods, the main focus has been put on their applicability and efficiency for data dependence analysis. Keywords: linear Diophantine equation, data dependence, data locality, dependence test, number theory 1 Introduction The extensive use of parallelism, fast processors and hierarchical memory systems greatly enhance the performance potential for modern architectures. However, compiler designers and programmers face the difficult task of making optimal use of these architectural improvements. One of the most crucial bottlenecks for the performance of..
