On the security of compressed encryption with partial unitary sensing matrices embedding a secret keystream

The principle of compressed sensing (CS) can be applied in a cryptosystem by providing the notion of security. In this paper, we study the computational security of a CS-based cryptosystem that encrypts a plaintext with a partial unitary sensing matrix embedding a secret keystream. The keystream is obtained by a keystream generator of stream ciphers, where the initial seed becomes the secret key of the CS-based cryptosystem. For security analysis, the total variation distance, bounded by the relative entropy and the Hellinger distance, is examined as a security measure for the indistinguishability. By developing upper bounds on the distance measures, we show that the CS-based cryptosystem can be computationally secure in terms of the indistinguishability, as long as the keystream length for each encryption is sufficiently large with low compression and sparsity ratios. In addition, we consider a potential chosen plaintext attack (CPA) from an adversary, which attempts to recover the key of the CS-based cryptosystem. Associated with the key recovery attack, we show that the computational security of our CS-based cryptosystem is brought by the mathematical intractability of a constrained integer least-squares (ILS) problem. For a sub-optimal, but feasible key recovery attack, we consider a successive approximate maximum-likelihood detection (SAMD) and investigate the performance by developing an upper bound on the success probability. Through theoretical and numerical analyses, we demonstrate that our CS-based cryptosystem can be secure against the key recovery attack through the SAMD.