Embedding Cryptographically Secure Matrix Transformation in Structured Compressive Sensing

In the context of secure communication using Compressive Sensing (CS) embedded Cryptography, the measurement matrix can be used as a secret key if it fulfills some requirements. This secret measurement matrix shall be generated at each communication entity using pre-shared information. For this reason, several measurement matrix generation mechanisms have been developed. In this paper we propose a mechanism for the design and the embedding of a cryptographically secure matrix transformation in structured CS. For this purpose, existing deterministic constructions of structured matrices, which are appropriate for specific applications, are used as input of the proposed mechanism. At the transmitter side, a secret matrix is generated using the designed secure matrix transformation. This matrix possesses the requirements of the corresponding CS-based application as well as the requirements of a cryptographic key. At the receiver side, the same secure matrix transformations are embedded into the CS reconstruction algorithm, so that the structure of the original non-secure measurement matrix can be reused during the recovery/decryption of the original signal. The proposed mechanism considers the fact that the Spark, the Null Space Property (NSP) order and the Restricted Isometry Property (RIP) order of CS measurement matrices, possess invariance under elementary column and row transformation of this matrices.