In recent years the synchronization of a chaotic response system with a chaotic drive system has attracted great interest in nonlinear science and engineering, due to the potential applications in chaos-based communications. This paper focuses on a new approach to achieve synchronization between two chaotic (hyperchaotic) maps with different dimensions. In particular, given a drive system map with dimension n and a response system with dimension m, the proposed approach enables each response system state to be synchronized with any linear combination of the drive system states. The method is based on a theorem that assures synchronization is achieved under certain broad conditions involving nilpotent matrices and suitable coupling between the two maps. The approach presents some new useful features, including the fact that exact synchronization up to an arbitrary scaling matrix is achievable in finite time for a wide class of chaotic maps with different dimensions. Moreover, the method is rigorous, systematic and readily implemented. The technique is illustrated by examples of synchronization between the three-dimensional case of the generalized Henon map and a recently introduced two-dimensional quadratic map. These examples highlight that exact synchronization is effectively achieved in finite time for any scaling matrix

Keywords: Chaotic maps, Exact synchronization in finite time, Attractor shaping