On the security of random linear measurements

In this paper, we analyze the security of compressed sensing (CS) as a cryptosystem. We demonstrate that random linear measurements acquired using a Gaussian i.i.d. matrix reveal only the energy of the sensed signal, and that only the energy of the measurements leaks information about the signal. We provide useful bounds for assessing the information leakage about the energy, linking those bounds to the minimum mean square error achievable by practical estimators. Moreover, we propose a simple strategy based on the normalization of the measurements which achieves, at least in theory, perfect secrecy, enabling the use of CS-based encryption in practical cryptosystems.