Security Aspects of Compressed Sensing

In this chapter, we will consider the security achievable by the compressed sensing (CS) framework under different constructions of the sensing matrix. CS can provide a form of data confidentiality when the signals are sensed by a random matrix composed of i.i.d. Gaussian variables. However, alternative constructions, based either on different distribution or on circulant matrices, which have similar CS recovery performance as Gaussian random matrices and admit faster implementations, are more suitable for practical CS systems. Compared to Gaussian matrices, which leak only the energy of the sensed signal, we show that generic matrices leak also some information about the structure of the sensed signal. In order to characterize this information leakage, we propose an operational definition of security linked to the difficulty of distinguishing equal energy signals and we propose practical attacks to test this definition. The results provide interesting insights on the security of generic sensing matrices, showing that a properly randomized partial circulant matrix can provide a weak encryption layer irrespective of the signal sparsity and the sensing domain.