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.
KeywordsDiscrete Fourier Transform Random Matrice Random Matrix Compress Sense Multivariate Gaussian Distribution
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 279848.
- 1.Bianchi T, Bioglio V, Magli E (2014) On the security of random linear measurements. In: 2014 IEEE International conference on acoustics, speech and signal processing (ICASSP’14), pp 3992–3996, doi: 10.1109/ICASSP.2014.6854351
- 2.Cambareri V, Haboba J, Pareschi F, Rovatti H, Setti G, Wong KW (2013) A two-class information concealing system based on compressed sensing. In: ISCAS’13, pp 1356–1359, doi: 10.1109/ISCAS.2013.6572106
- 5.Cover TM, Thomas JA (2006) Elements of Information Theory. Wiley-Interscience, HobokenGoogle Scholar
- 13.Hershey J, Olsen P (2007) Approximating the Kullback Leibler divergence between Gaussian mixture models. In: ICASSP’07, vol 4, pp IV-317–IV-320, doi: 10.1109/ICASSP.2007.366913
- 16.Orsdemir A, Altun H, Sharma G, Bocko M (2008) On the security and robustness of encryption via compressed sensing. In: IEEE Military communications conference, 2008 (MILCOM 2008), pp 1–7, doi: 10.1109/MILCOM.2008.4753187
- 17.Rachlin Y, Baron D (2008) The secrecy of compressed sensing measurements. In: IEEE 2008 46th Annual allerton conference on communication, control, and computing, pp 813–817, doi: 10.1109/ALLERTON.2008.4797641
- 18.Rauhut H (2009) Circulant and toeplitz matrices in compressed sensing. In: SPARS’09—Signal processing with adaptive sparse structured representationsGoogle Scholar
- 20.Valsesia D, Magli E (2014) Compressive signal processing with circulant sensing matrices. In: IEEE ICASSP’14, pp 1015–1019, doi: 10.1109/ICASSP.2014.6853750
- 21.Yin W, Morgan S, Yang J, Zhang Y (2010) Practical compressive sensing with Toeplitz and circulant matrices. In: Proceeding of SPIE, vol 7744, pp 77,440K–77,440K–10, doi: 10.1117/12.863527