To improve the capability of stereo image zero watermarking scheme anti-geometric attacks, this work a stereo image zero watermarking algorithm based on Fast Finite Shearlet Transform (FFST) and Hessenberg decomposition is proposed. Firstly, the FFST is applied to the luminance components on the left and right viewpoint respectively of the original stereo image in the YCbCr color space to obtain the two low-frequency subbands. Then the robust feature matrix is constructed based on the random sub-block selecting strategy and Hessenberg joint decomposition from the above subbands. Finally, the certified zero watermark is produced by executing exclusive OR on the robust matrix and the preprocessed binary watermark. Furthermore, the Fourier-Mellin Transform is used to correct the stereo image to be certified before zero watermarking detection. Compared with related algorithms, the algorithm in this paper has an average NC value of 0.984 1 for common non-geometric and geometric attacks, showing better anti-attack capability. The proposed algorithm has lossless and strong security, and can provide a feasible solution for stereo image copyright protection.