Indistinguishability and Energy Sensitivity of Gaussian and Bernoulli Compressed Encryption

The principle of compressed sensing (CS) can be applied in a cryptosystem by providing the notion of security. The Gaussian one-time sensing (G-OTS) CS-based cryptosystem employing a random Gaussian matrix and renewing the elements at each encryption is known to be perfectly secure, as long as each plaintext has constant energy. A random Bernoulli matrix can replace the Gaussian one for encrypting each plaintext efficiently in the Bernoulli one-time sensing (B-OTS) cryptosystem. In this paper, we analyze the security of G-OTS and B-OTS cryptosystems, respectively, where each cryptosystem may have unequal plaintext energy. By means of probability metrics, we study the indistinguishability of each CS-based cryptosystem, which formalizes the notion of computational security. Moreover, we investigate how much the indistinguishability is sensitive to energy variation of plaintexts in each cryptosystem. For the B-OTS cryptosystem, we analyze the indistinguishability and the energy sensitivity in a non-asymptotic manner for a finite plaintext length. In conclusion, this paper confirms that G-OTS and B-OTS cryptosystems can be strictly and asymptotically indistinguishable, respectively, as long as each plaintext has constant energy, but the indistinguishability is highly sensitive to energy variation of plaintexts.