In this paper, we propose L0 norm optimization in a scrambled sparse representation domain and its application to an Encryption-then-Compression (EtC) system. We design a random unitary transform that conserves L0 norm isometry. The resulting encryption method provides a practical orthogonal matching pursuit (OMP) algorithm that allows computation in the encrypted domain. We prove that the proposed method theoretically has exactly the same estimation performance as the nonencrypted variant of the OMP algorithm. In addition, we demonstrate the security strength of the proposed secure sparse representation when applied to the EtC system. Even if the dictionary information is leaked, the proposed scheme protects the privacy information of observed signals.