The scale invariant feature transform (SIFT) algorithm, commonly used in computer vision, does not perform well on synthetic aperture radar (SAR) images, in particular because of the strong intensity and the multiplicative nature of the noise. We present an improvement of this algorithm for SAR images. First, a robust yet simple way to compute gradient on radar images is introduced. This step is first used to develop a new keypoints extraction algorithm, based on the Harris criterion. Second, we rely on this gradient definition to adapt the computation of both the main orientation and the geometric descriptor to SAR image specificities. We validate this new algorithm with different experiments and present an application of our new SAR-SIFT algorithm.