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5 - Secrecy Rate Maximization in Gaussian MIMO Wiretap Channels

from Part II - Secure Communication

Published online by Cambridge University Press:  28 June 2017

By
S. Loyka  and
C. D. Charalambous
Edited by
Rafael F. Schaefer ,
Holger Boche ,
Ashish Khisti  and
H. Vincent Poor
S. Loyka
Affiliation:
School of Electrical Engineering and Computer Science, University of Ottawa
C. D. Charalambous
Affiliation:
Department of Electrical and Computer Engineering, University of Cyprus
Rafael F. Schaefer
Affiliation:
Technische Universität Berlin
Holger Boche
Affiliation:
Technische Universität München
Ashish Khisti
Affiliation:
University of Toronto
H. Vincent Poor
Affiliation:
Princeton University, New Jersey
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Summary

Secrecy rate maximization in Gaussian MIMO wiretap channels is considered. While the optimality of Gaussian signaling and a general expression for the secrecy capacity have been well established, closed-form solutions for the optimal transmit covariance matrix are known for some special cases only, while the general case remains an open problem. This chapter reviews known closed-form solutions and presents a numerical algorithm for the general case with guaranteed convergence to the global optimum. The known solutions include full-rank and rank-1 cases (which, when combined, provide a complete solution for the case of two transmit antennas), the case of identical right singular vectors for the eavesdropper and legitimate channels, and the cases of weak, isotropic, and omnidirectional eavesdroppers, which also provide lower and upper bounds to the general case. Necessary optimality conditions and a tight upper bound for the rank of the optimal covariance matrix in the general case are discussed. Sufficient and necessary conditions for the optimality of three popular signaling strategies over MIMO channels, namely, isotropic and zero-forcing signaling as well as water-filling over the legitimate channel eigenmodes, are presented. The chapter closes with a detailed description of a numerical globally convergent algorithm to solve the general case, and gives some illustrative examples.

Introduction

Due to their high spectral efficiency, wireless MIMO (multiple input, multiple output) systems are widely adopted by academia and industry. The broadcast nature of wireless channels stimulated significant interest in their security aspects and the Gaussian MIMO wiretap channel (WTC) has emerged as a popular model to study information theoretic secrecy aspects of wireless systems [1]. A number of results have been obtained for this model, including the proof of optimality of Gaussian signaling [1–4], which is far from trivial and significantly more involved than that of the regular (no wiretap) MIMO channel. Once the functional form of the optimal input is established, the only unknown is its covariance matrix since the mean is always zero. This latter part has not been solved yet in the general case; only a number of special cases have been settled.

Type
Chapter
Information
Information Theoretic Security and Privacy of Information Systems , pp. 109 - 139
Publisher: Cambridge University Press
Print publication year: 2017

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References

[1] M., Bloch and J., Barros, Physical-Layer Security: From Information Theory to Security Engineering. Cambridge: Cambridge University Press, 2011.
[2] A., Khisti and G. W., Wornell, “Secure transmission with multiple antennas i: The MISOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3088–3104, Jul. 2010.Google Scholar
[3] A., Khisti and G. W., Wornell, “Secure transmission with multiple antennas–Part II: The MIMOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5515–5532, Nov. 2010.Google Scholar
[4] F., Oggier and B., Hassibi, “The secrecy capacity of theMIMO wiretap channel,” IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 4961–4972, Aug. 2011.Google Scholar
[5] T., Liu and S., Shamai (Shitz), “A note on the secrecy capacity of the multiple-antenna wiretap channel,” IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2547–2553, Jun. 2009.Google Scholar
[6] Z., Li, W., Trappe, and R., Yates, “Secret communication via multi-antenna transmission,” in Proc. Conf. Inf. Sciences Systems, Baltimore, MD, USA, Mar. 2007, pp. 905–910.
[7] J., Li and A., Petropulu, “Transmitter optimization for achieving secrecy capacity in Gaussian MIMO wiretap channels,” Sep. 2009. [Online]. Available: http://arxiv.org/abs/0909.2622
[8] S., Shafiee, N., Liu, and S., Ulukus, “Towards the secrecy capacity of the Gaussian MIMO wire-tap channel: The 2-2-1 channel,” IEEE Trans. Inf. Theory, vol. 55, no. 9, pp. 4033–4039, Sep. 2009.Google Scholar
[9] M. C., Gursoy, “Secure communication in the low-SNR regime,” IEEE Commun., vol. 60, no. 4, pp. 1114–1123, Apr. 2012.Google Scholar
[10] S., Loyka and C. D., Charalambous, “On optimal signaling over secure MIMO channels,” in Proc. IEEE Int. Symp. Inf. Theory, Cambridge, MA, USA, Jul. 2012, pp. 443–447.
[11] S., Loyka and C. D., Charalambous, “Further results on optimal signaling over secureMIMO channels,” in Proc. IEEE Int. Symp. Inf. Theory, Istanbul, Turkey, Jul. 2013, pp. 2019–2023.
[12] S., Loyka and C. D., Charalambous, “Rank-deficient solutions for optimal signaling over secure MIMO channels,” in Proc. IEEE Int. Symp. Inf. Theory, Honolulu, HI, USA, Jun. 2014, pp. 201–205.
[13] S., Loyka and C. D., Charalambous, “An algorithm for global maximization of secrecy rates in Gaussian MIMO wiretap channels,” IEEE Trans. Commun., vol. 63, no. 6, pp. 2288–2299, Jun. 2015.Google Scholar
[14] A., Khisti, “Interference alignment for the multiantenna compound wiretap channel,” IEEE Trans. Inf. Theory, vol. 57, no. 5, pp. 2976–2993, May 2011.
[15] I., Bjelaković, H., Boche, and J., Sommerfeld, “Secrecy results for compound wiretap channels,” Probl. Inf. Transmission, vol. 49, no. 1, pp. 73–98, Mar. 2013.Google Scholar
[16] R. F., Schaefer and S., Loyka, “The secrecy capacity of compound MIMO Gaussian channels,” IEEE Trans. Inf. Theory, vol. 61, no. 10, pp. 5535–5552, Dec. 2015.Google Scholar
[17] R. A., Horn and C. R., Johnson, Matrix Analysis. Cambridge: Cambridge University Press, 1999.
[18] T. S., Rappaport, Wireless Communications, 2nd edn. Upper Saddle River, NJ: Prentice Hall, 2002.
[19] S., Loyka and G., Levin, “On physically-based normalization of MIMO channel matrices,” IEEE Trans. Wireless Commun., vol. 8, no. 3, pp. 1107–1112, Mar. 2009.Google Scholar
[20] A., Khisti, A., Tchamkerten, and G.W., Wornell, “Secure broadcasting over fading channels,” IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2453–2469, Jun. 2008.Google Scholar
[21] Z., Li, R., Yates, and W., Trappe, Securing Wireless Communications at the Physical Layer. Boston, MA: Springer US, 2010, pp. 1–18.
[22] D. N. C., Tse and P., Viswanath, Fundamentals of Wireless Communication. Cambridge: Cambridge University Press, 2005.
[23] Z., Rezki, A., Khisti, and M.-S., Alouini, “Ergodic secret message capacity of the wiretap channel with finite-rate feedback,” IEEE Trans. Wireless Commun., vol. 13, no. 6, pp. 3364–3379, Jun. 2014.Google Scholar
[24] J. P., Kermoal, L., Schumacher, K. I., Pedersen, P. E., Mogensen, and F., Frederiksen, “A stochastic MIMO radio channel model with experimental validation,” IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. 1211–1226, Jun. 2002.Google Scholar
[25] H. L. van, Trees, Optimum Array Processing. Chichester: Wiley, 2002.
[26] T. M., Cover and J. A., Thomas, Elements of Information Theory, 2nd edn. Chichester:Wiley & Sons, 2006.
[27] Q., Li, M., Hong, H.-T., Wai, Y.-F., Liu, W.-K., Ma, and Z.-Q., Luo, “Transmit solutions for MIMO wiretap channels using alternating optimization,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 1714–1727, Sep. 2013.Google Scholar
[28] J., Steinwandt, S. A., Vorobyov, and M., Haardt, “Secrecy rate maximization for MIMO Gaussian wiretap channels with multiple eavesdroppers via alternating matrix POTDC,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Process., Florence, Italy, May 2014, pp. 5686–5690.
[29] S. P., Boyd and L., Vandenberghe, Convex Optimization. Cambridge: Cambridge University Press, 2004.

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