In this paper, we propose a secure computation of sparse coding and its application to Encryption-then-Compression (EtC) systems. The proposed scheme introduces secure sparse coding that allows computation of an Orthogonal Matching Pursuit (OMP) algorithm in an encrypted domain. We prove theoretically that the proposed method estimates exactly the same sparse representations that the OMP algorithm for non-encrypted computation does. This means that there is no degradation of the sparse representation performance. Furthermore, the proposed method can control the sparsity without decoding the encrypted signals. Next, we propose an EtC system based on the secure sparse coding. The proposed secure EtC system can protect the private information of the original image contents while performing image compression. It provides the same rate-distortion performance as that of sparse coding without encryption, as demonstrated on both synthetic data and natural images.