This paper introduces a novel technique to automatically recompose archaeological broken artifacts from fragments. The aim of the proposed technique is to assist restoration personnel in the tedious task of reconstructing ancient relics found in excavation sites. Starting from unstructured point clouds of fragments, our approach attempts to recompose the original artifact by finding the individual rigid transformations that better align all the fragments. To efficiently do so, a pre-processing stage executed where the size of the problem is considerably reduced: singular keypoints in the original point cloud are selected based on a multi-scale feature extraction process driven by the saliency of each point. Computing a modified version of the PFH descriptor, the local neighborhood of each keypoint is described in a compact histogram. Using exclusively the selected keypoints and their associated descriptors, a very fast one-to-one search algorithm is executed for each possible pair of fragments. This process uses a three-level hierarchical search strategy driven by the local similarity between keypoints, and applying a set of geometric consistence tests for intermediate results. Finally, a graph-based global registration algorithm uses all the individual matches to compute the final reconstruction of the artifact by creating clusters of matching fragments, appending new potential matches and joining individual clusters into bigger structures. Empirical tests have proven the technique to be extremely fast while keeping memory requirements very low.