How can we solve a linear Diophantine equation by the basis reduction algorithm

Abstract

The main difficulty in solving the systems of linear Diophantine equations is the very rapid growth of the intermediate results. Hence, it is important to design algorithms that restraint the growth of intermediate results. The use of the basis reduction algorithm for solving a linear Diophantine equation is examined in this paper. The motivation behind choosing the basis reduction is twofold. First the basis reduction allows us to work only with integers, which avoids the round-off problems. Second, the basis reduction finds short and nearly orthogonal vectors belonging to the lattice described by the basis. Then, we expect the general solution obtained by this basis will also be short. It is important to note that the basis reduction does not change the lattice, it only derives an alternative way of spanning it. Once we have obtained the vectors given by the reduced basis, we use them to find the general solution of the linear Diophantine equation.

Keywords:

• Linear Diophantine equation
• Polynomial time algorithm
• Unimodular matrices
• Basis reduction algorithm