A compressed sensing-based progressive secret image sharing scheme and security analysis

Abstract

Cryptographic secret image sharing (SIS) allows a secret image to be shared among participants, and any authorized set of participants can reconstruct the secret. In this paper, we first propose a conceptually simple way to construct a progressive SIS scheme based on semi-tensor product compressed sensing (STP-CS). In the sharing phase, the dealer employs STP-CS to compress the secret image and extracts several different rows of the compressed image in turn. Then the sub-shadows for participants are formed. In addition, he needs to employ a perfect secret sharing to share the measurement matrix (or an initial state of the linear feedback shift register) among all participants. Finally, the sub-shadow and the share of the measurement matrix together constitute the shadow of each participant. In the combination phase, any legitimate access structure can reconstruct the secret image gradually. The perfect secret sharing ensures that the combination phase can work if and only if the access structure is authorized. Simultaneously, the use of STP-CS ensures that the proposed scheme has the properties of gradual reconstruction, noise-resilience, and smaller shadow size. Also, we have proven that the proposed scheme achieves the extended Wyner-sense perfect secrecy. Besides, we provide an extend version, referred to as homomorphic secret image sharing (HSIS), for improving the efficiency of the combination algorithm. HSIS can outsource the combination phase to the cloud servers securely.