Solving Sparse Integer Linear Systems
  We propose a new algorithm to solve sparse linear systems of equations over the integers. This algorithm is based on a $p$-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art. 
Wayne Eberly, Mark Giesbrecht (SCG), Pascal Giorgi (LP2A, SCG), Arne   Storjohann (SCG), Gilles Villard (LIP)