Security Aspects of Compressed Sensing

  • Tiziano BianchiEmail author
  • Enrico Magli
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 358)


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.


Discrete Fourier Transform Random Matrice Random Matrix Compress Sense Multivariate Gaussian Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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. 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. 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
  3. 3.
    Candes E, Tao T (2006) Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans Inf Theory 52(12):5406–5425. doi: 10.1109/TIT.2006.885507 zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Candes E, Romberg J, Tao T (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52(2):489–509. doi: 10.1109/TIT.2005.862083 zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Cover TM, Thomas JA (2006) Elements of Information Theory. Wiley-Interscience, HobokenGoogle Scholar
  6. 6.
    Do M (2003) Fast approximation of Kullback-Leibler distance for dependence trees and hidden Markov models. IEEE Signal Process Lett 10(4):115–118. doi: 10.1109/LSP.2003.809034 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Do T, Gan L, Nguyen N, Tran T (2012) Fast and efficient compressive sensing using structurally random matrices. IEEE Trans Signal Process 60(1):139–154. doi: 10.1109/TSP.2011.2170977 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Donoho D (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306. doi: 10.1109/TIT.2006.871582 zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fanzi Z, Li C, Tian Z (2011) Distributed compressive spectrum sensing in cooperative multihop cognitive networks. IEEE J Sel Topics Signal Process 5(1):37–48. doi: 10.1109/JSTSP.2010.2055037 CrossRefGoogle Scholar
  10. 10.
    Goldwasser S, Micali S (1984) Probabilistic encryption. J Comput Syst Sci 28(2):270–299. doi: 10.1016/0022-0000(84)90070-9 zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Haupt J, Bajwa W, Rabbat M, Nowak R (2008) Compressed sensing for networked data. IEEE Signal Process Mag 25(2):92–101. doi: 10.1109/MSP.2007.914732 CrossRefGoogle Scholar
  12. 12.
    Haupt J, Bajwa W, Raz G, Nowak R (2010) Toeplitz compressed sensing matrices with applications to sparse channel estimation. IEEE Trans Inf Theory 56(11):5862–5875MathSciNetCrossRefGoogle Scholar
  13. 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
  14. 14.
    Lehmann EL, Romano JP (2005) Testing Statistical Hypotheses, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
  15. 15.
    Mardani M, Mateos G, Giannakis G (2013) Dynamic anomalography: tracking network anomalies via sparsity and low rank. IEEE J Sel Topics Signal Process 7(1):50–66. doi: 10.1109/JSTSP.2012.2233193 CrossRefGoogle Scholar
  16. 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. 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. 18.
    Rauhut H (2009) Circulant and toeplitz matrices in compressed sensing. In: SPARS’09—Signal processing with adaptive sparse structured representationsGoogle Scholar
  19. 19.
    Shannon CE (1949) Communication theory of secrecy systems. Bell Syst Tech J 28:656–715zbMATHMathSciNetCrossRefGoogle Scholar
  20. 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. 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

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Politecnico di TorinoTorinoItaly

Personalised recommendations