Copy–move forgery is one of the most frequently used methods for producing fake digital images. Current algorithms for copy–move forgery detection (CMFD) cannot combine high accuracy and fast speed. Motivated by the observation, we propose a novel CMFD algorithm whose workflow is as follows. First, we use a keypoint-extraction method with the lowest contrast threshold to extract more keypoints from the input image. Second, a new technique, gradient-hash matching, finds pairs of similar keypoints quickly and effectively using a hash table, where the hash value is computed using gradients of keypoints. Subsequently, a new method called simplified cluster-based filtering exploits the density pattern of keypoints in the copy–move regions to remove false matching keypoint pairs. Finally, image matting is applied to indicate the forgery regions vividly. Extensive experiments show that not only the new algorithm is better than the state-of-the-art algorithms in terms of computation correctness, but also its computation time is drastically less. Commonly only about half time is needed. The relative time saving is even higher when images are larger. Different algorithms modules are compared through experiments to choose the best combination.