Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T21:39:17.829Z Has data issue: false hasContentIssue false

9 - Stochastic Orders, Alignments, and Ergodic Secrecy Capacity

from Part II - Secure Communication

Published online by Cambridge University Press:  28 June 2017

By
P.-H. Lin  and
E. A. Jorswieck
Edited by
Rafael F. Schaefer ,
Holger Boche ,
Ashish Khisti  and
H. Vincent Poor
P.-H. Lin
Affiliation:
Chair for Communications Theory, Technische Universität Dresden
E. A. Jorswieck
Affiliation:
Chair for Communications Theory, Technische Universität Dresden
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
Get access

Summary

We investigate the relation between different stochastic orders and the degradedness of a fast fading wiretap channel with statistical channel state information at the transmitter (CSIT). In particular, we derive sufficient conditions to identify the ergodic secrecy capacities for both single and multiple antenna cases even though there is only statistical CSIT.

Introduction

To design a wiretap code with higher secrecy rate, channel state information at the transmitter of the legitimate and eavesdropper's channels1 should be known to a certain degree. When there is perfect CSIT of both channels, the secrecy capacity of a Gaussian wiretap channel can be achieved by a Gaussian input. However, due to several practical issues such as (1) being a malicious user, Eve is by no means intending to feed back the correct CSI to Alice; (2) limited feedback bandwidth; (3) the delay caused by channel estimation; (4) the speed of channel variation, etc., it is more reasonable to consider cases where perfect CSIT is unavailable.

In this chapter we consider the cases when only statistical CSIT of both channels from Alice to Bob and Alice to Eve, respectively, are available. One possible way to get such information from Eve virtually is as follows. For some bounded space like indoor parking lots or malls, it is possible to collect the channel statistics offline of most of the positions within that space, where the channel variation may be due to the movement of people or cars, etc. When Alice wants to transmit, she can use the known statistics measured offline of the channel between her and the unknown user closest to her as Eve's channel information to design her wiretap code for the worst-case scenario.

Although the secrecy capacity formula for discrete memoryless non-degraded wiretap channels was proved in [1], the optimal selection of the auxiliary random variable and channel prefixing are still unknown for additive white Gaussian noise (AWGN) channels with partial CSIT in general. Only a few capacity results under the considered scenario are known, and are summarized as follows.

Type
Chapter
Information
Information Theoretic Security and Privacy of Information Systems , pp. 231 - 257
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] I., Csiszár and J., Körner, “Broadcast channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 24, no. 3, pp. 339–348, May 1978.Google Scholar
[2] S.-C., Lin and P.-H., Lin, “On ergodic secrecy capacity of multiple input wiretap channel with statistical CSIT,” IEEE Trans. Inf. Forensics Security, vol. 8, no. 2, pp. 414–419, Feb. 2013.Google Scholar
[3] S.-C., Lin and C.-L., Lin, “On secrecy capacity of fast fading MIMOME wiretap channels with statistical CSIT,” IEEE Trans. Wireless Commun., vol. 13, no. 6, pp. 3293–3306, Jun. 2014.Google Scholar
[4] P., Mukherjee and S., Ulukus, “Fading wiretap channel with no CSI anywhere,” in Proc. IEEE Int. Symp. Inf. Theory, Istanbul, Turkey, Jul. 2013, pp. 1347–1351.
[5] S., Goel and R., Negi, “Guaranteeing secrecy using artificial noise,” IEEE Trans. Wireless Commun., vol. 7, no. 6, pp. 2180–2189, Jun. 2008.Google Scholar
[6] J., Li and A., Petropulu, “On ergodic secrecy rate for Gaussian MISO wiretap channels,” IEEE Trans. Wireless Commun., vol. 10, no. 4, pp. 1176–1187, Apr. 2011.Google Scholar
[7] M. R., Bloch and J. N., Laneman, “Exploiting partial channel state information for secrecy over wireless channels,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 1840–1849, Sep. 2013.Google Scholar
[8] P.-H., Lin, S.-H., Lai, S.-C., Lin, and H.-J., Su, “On Secrecy Rate of the Generalized Artificial-Noise Assisted Secure Beamforming for Wiretap Channels,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 1728–1740, Sep. 2013.Google Scholar
[9] 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
[10] Y., Liang, H. V., Poor, and S., Shamai (Shitz), “Information theoretic security,” Found. Trends Commun. Inf. Theory, vol. 5, no. 4–5, pp. 355–580, 2009.Google Scholar
[11] M., Shaked and J. G., Shanthikumar, Stochastic Orders. New York: Springer, 2007.
[12] B., Hajek, Notes for ECE 534: An Exploration of Random Processes for Engineers. 2014. [Online]. Available: www.ifp.illinois.edu/hajek/Papers/randomprocJan14.pdf
[13] S. M., Ross, Stochastic Process, 2nd edn. New York: John Wiley & Sons, 1996.
[14] H., Colonius, “An invitation to coupling and copulas: With applications to multisensory modeling,” Nov. 2015. [Online]. Available: http://arxiv.org/abs/1511.05303
[15] G., Caire and S., Shamai (Shitz), “On the capacity of some channels with channel state information,” IEEE Trans. Inf. Theory, vol. 45, no. 6, pp. 2007–2019, Sep. 1999.Google Scholar
[16] A. El, Gamal and Y.-H., Kim, Network Information Theory. Cambridge: Cambridge University Press, 2011.
[17] J., Körner and K., Marton, “Comparison of two noisy channels,” in Colloquia Mathematica Societatis, Jańos Bolyai, 16, Topics in Info. Th., North Holland, 1977, pp. 411–424.
[18] P.-H., Lin and E. A., Jorswieck, “On the fading Gaussian wiretap channel with statistical channel state information at transmitter,” IEEE Trans. Inf. Forensics Security, vol. 11, no. 1, pp. 46–58, Jan. 2016.Google Scholar
[19] D. N. C., Tse and R. D., Yates, “Fading broadcast channels with state information at the receivers,” IEEE Trans. Inf. Theory, vol. 58, no. 6, pp. 3453–3471, Jun. 2012.Google Scholar
[20] A., Rajan and C., Tepedelenlioglu, “Stochastic ordering of fading channels through the Shannon transform,” IEEE Trans. Inf. Theory, vol. 61, no. 4, pp. 1619–1628, Apr. 2015.Google Scholar
[21] Z., Li, R., Yates, and W., Trappe, “Achieving secret communication for fast Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 9, no. 9, pp. 2792–2799, Sep. 2010.Google Scholar
[22] A. S., Avestimehr, S., Diggavi, and D., Tse, “Wireless network information flow: A deterministic approach,” IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 1872–1905, Apr. 2011.Google Scholar
[23] Y., Liang, H. V., Poor, and S., Shamai (Shitz), “Secure communication over fading channels,” IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2470–2492, Jun. 2008.Google Scholar
[24] A., Martinez, “Communication by energy modulation: The additive exponential noise channel,” IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3333–3351, Jun. 11.Google Scholar
[25] O. O., Koyluoglu, K., Appaiah, and S., Vishwanath, “Expansion coding: Achieving the capacity of an AEN channel,” in Proc. IEEE Int. Symp. Inf. Theory, Cambridge, MA, USA, Jul. 2012, pp. 1932–1936.
[26] E., Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051–3073, Jul. 2009.Google Scholar
[27] P.-H., Lin, E. A., Jorswieck, R. F., Schaefer, and M., Mittelbach, “On the degradedness of fast fading Gaussian multiple-antenna wiretap channels with statistical channel state information at the transmitter,” in Proc. IEEE Global Commun. Conf. Workshops, SanDiego, CA, USA, Dec. 2015, pp. 1–5.
[28] Z., Sasvari, Multivariate Characteristic and Correlation Functions. Berlin: De Gruyter Studies in Mathematics, 2013.
[29] G. A. F., Seber, A Matrix Handbook for Statisticians. Hoboken, NJ: Wiley Series in Probability and Statistics, 2008.
[30] A. W., Marshall and I., Olkin, Inequalities: Theory of Majorization and its Application, 2nd edn. London: Academic Press, 1979.
[31] S., Jin, M. R., McKay, X., Gao, and I. B., Collings, “MIMO multichannel beamforming: SER and outage using new eigenvalue distributions of complex noncentral Wishart matrices,” IEEE Trans. Commun., vol. 56, no. 3, pp. 424–434, Mar. 2008.Google Scholar
[32] A. H., Nuttall, “Some integrals involving the Q-function,” Naval Underwater Systems Center, New London Lab., New London, CT, Tech. Rep. 4297, Apr. 1972.
[33] 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.
[34] J., Li and A., Petropulu, “Optimality of beamforming and closed form secrecy capacity of MIMO wiretap channels with two transmit antennas,” in Proc. IEEE 16th Int. Symp. Wireless Personal Multimedia Commun., Jun. 2013, pp. 1–5.Google Scholar
[35] S. A. A., Fakoorian and A. L., Swindlehurst, “Full rank solutions for the MIMO Gaussian wiretap channel with an average power constraint,” IEEE Trans. Signal Process., vol. 61, no. 10, pp. 2620–2631, May 2013.Google Scholar
[36] A., Gupta and D., Nagar, Matrix Variate Distributions. London: CRC Press, 2000.
[37] B., Hassibi and T. L., Marzetta, “Multiple antennas and isotropically random unitary inputs: The received signal density in closed form,” IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1473–1484, Jun. 2002.Google Scholar
[38] S., Loyka and C.D., Charalambos, “An Algorithm for Global Maximization of Secrecy Rates in Gaussian MIMO Wiretap Channels,” IEEE Trans. Commun.,vol. 63, no.6, pp. 2288–12299, 2015.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×