Analysis of One-Time Random Projections for Privacy Preserving Compressed Sensing

In this paper, the security of the compressed sensing (CS) framework as a form of data confidentiality is analyzed. Two important properties of one-time random linear measurements acquired using a Gaussian independent identically distributed matrix are outlined: 1) the measurements reveal only the energy of the sensed signal and 2) only the energy of the measurements leaks information about the signal. An important consequence of the above facts is that CS provides information theoretic secrecy in a particular setting. Namely, a simple strategy based on the normalization of the Gaussian measurements achieves, at least in theory, perfect secrecy, enabling the use of CS as an additional security layer in privacy preserving applications. In the generic setting in which CS does not provide information theoretic secrecy, two alternative security notions linked to the difficulty of estimating the energy of the signal and distinguishing equal-energy signals are introduced. Useful bounds on the mean square error of any possible estimator and the probability of error of any possible detector are provided and compared with the simulations. The results indicate that CS is in general not secure according to cryptographic standards, but may provide a useful built-in data obfuscation layer.