Detecting reliably copy-move forgeries is difficult because images do contain similar objects. The question is: how to discard natural image self-similarities while still detecting copy-moved parts as being "unnaturally similar"? Copy-move may have been performed after a rotation, a change of scale and followed by JPEG compression or the addition of noise. For this reason, we base our method on SIFT, which provides sparse keypoints with scale, rotation and illumination invariant descriptors. To discriminate natural descriptor matches from artificial ones, we introduce an a contrario method which gives theoretical guarantees on the number of false alarms. We validate our method on several databases. Being fully unsupervised it can be integrated into any generic automated image tampering detection pipeline.