Two algorithms for the solution of linear Diophantine systems, which well restrain intermediate expression swell, are presented. One is an extension and improvement of Kannan and Bachem’s algorithm for the Smith and the Hermite normal forms of a nonsingular square integral matrix. The complexity of this algorithm is investigated and a polynomial time bound is derived. Also a much better coefficient bound is obtained compared to Kannan and Bachem’s analysis. The other is based on ideas of Rosser, which were originally used in finding a general solution with smaller coefficients to a linear Diophantine equation and in computing the exact inverse of a nonsingular square integral matrix. These algorithms are implemented by using the infinite precision integer arithmetic capabilities of the SAC-2 system. Their performances are compared. Finally future studies are mentioned.