In this paper, based on second-order cross-partial derivative (CPD), we propose an efficient blind image deblurring algorithm for uniform blur. The proposed method consists of two stages. We first apply a novel blur kernel estimation method to quickly estimate the blur kernel. Then, we use the estimated kernel to perform non-blind deconvolution to restore the image. A key discovery of the proposed kernel estimation method is that the blur kernel information is usually embedded in the cross-partial-derivative (CPD) image of the blurred image. By exploiting this property, we propose a pipeline to extract a set of kernel candidates directly from the CPD image and then select the most suitable kernel as the estimated blur kernel. Since our kernel estimation method can obtain a fairly accurate blur kernel, we can achieve effective image restoration using a relatively simple Tikhonov regularization in the subsequent non-blind deconvolution process. To improve the quality of the restored image, we further adopt an efficient filtering technique to suppress periodic artifacts that may appear in the restored images. Experimental results demonstrate that our algorithm can efficiently restore high-quality sharp images on standard CPUs without relying on GPU acceleration or parallel computation. For blurred images of approximately $800\times 800$ resolution, the proposed method can complete image deblurring within 1 to 5 seconds, which is significantly faster than most state-of-the-art methods. Our MATLAB codes are available at https://github.com/e11tkcee06-a11y/CPD-Deblur.git.