Skip to main content Accessibility help
  • Print publication year: 2017
  • Online publication date: June 2017

11 - Super-Activation as a Unique Feature of Secure Communication over Arbitrarily Varying Channels

from Part II - Secure Communication


The question of whether the capacity of a channel is additive or not goes back to Shannon, who asked this for the zero-error capacity function. Despite the common sense that the capacity is usually additive, there is surprisingly little known for non-trivial channels. This chapter addresses this question for the arbitrarily varying wiretap channel (AVWC), which models secure communication in the presence of arbitrarily varying channel (AVC) conditions. For orthogonal AVWCs it has been shown that the strongest form of non-additivity occurs: the phenomenon of super-activation. That is, there are orthogonal AVWCs, each having zero secrecy capacity, which allow for transmission with positive rate if they are used together. Subsequently, the single-user AVC is studied and it is shown that in this case, super-activation for non-secure message transmission is not possible, making it a unique feature of secure communication over AVWCs. However, the capacity for message transmission of the single-user AVC is shown to be super-additive, including a complete characterization. Super-activation was known for a long time in the area of quantum communication, where it is common opinion that this is solely a phenomenon of quantum physics and that this cannot occur for classical communication. However, the results in this chapter show that super-activation is indeed a feature of secure communication and therewith occurs in classical, non-quantum, communication as well.


Information theory goes back to Shannon's seminal work “A Mathematical Theory of Communication” [1]. Since then it has been proven to be an indispensable concept for analyzing complex communication systems to obtain insights and optimal design criteria. Among many things, it is used to address the important issue of medium access control: How should available resources be divided among multiple users in the best possible way? This is a crucial question, especially for wireless communication systems since they are usually composed of orthogonal sub-systems such as those that arise via time division multiplexing (TDM) or orthogonal frequency division multiplexing (OFDM). Of particular interest then is to know how the capacity of the overall system depends on the orthogonal sub-systems. Common sense tells us that for such systems it should be given by the sum of the capacities of all sub-systems. This goes along with the inherent world view of the additivity of classical resources.

[1] C. E., Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423, 623–656, Jul., Oct. 1948.
[2] C. E., Shannon, “The zero error capacity of a noisy channel,” IRE Trans. Inf. Theory, vol. 2, no. 3, pp. 8–19, Sep. 1956.
[3] L., Lovász, “On the Shannon capacity of a graph,” IEEE Trans. Inf. Theory, vol. 25, no. 1, pp. 1–7, Jan. 1979.
[4] W., Haemers, “On some problems of Lovász concerning the Shannon capacity of a graph,” IEEE Trans. Inf. Theory, vol. 25, no. 2, pp. 231–232, Mar. 1979.
[5] N., Alon, “The Shannon capacity of a union,” Combinatorica, vol. 18, no. 3, pp. 301–310, Mar. 1998.
[6] R., Ahlswede, “A note on the existence of the weak capacity for channels with arbitrarily varying channel probability functions and its relation to Shannon's zero error capacity,” Ann. Math. Stat., vol. 41, no. 3, pp. 1027–1033, 1970.
[7] A. D., Wyner, “The wire-tap channel,” Bell Syst. Tech. J., vol. 54, pp. 1355–1387, Oct. 1975.
[8] 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.
[9] R., Liu and W., Trappe, eds., Securing Wireless Communications at the Physical Layer. Boston, MA: Springer US, 2010.
[10] M., Bloch and J., Barros, Physical-Layer Security: From Information Theory to Security Engineering. Cambridge: Cambridge University Press, 2011.
[11] H. V., Poor and R. F., Schaefer, “Wireless physical layer security,” Proc. Natl. Acad. Sci. U.S.A., vol. 114, no. 1, pp. 19–26, Jan. 3, 2017.
[12] R. F., Schaefer and H., Boche, “Physical layer service integration in wireless networks – signal processing challenges,” IEEE Signal Process. Mag., vol. 31, no. 3, pp. 147–156, May 2014.
[13] U., Helmbrecht and R., Plaga, “New challenges for IT-security research in ICT,” in World Federation of Scientists International Seminars on Planetary Emergencies, Erice, Italy, Aug. 2008, pp. 1–6.
[14] Deutsche Telekom AG Laboratories, “Next generation mobile networks: (R)evolution in mobile communications,” Technology Radar Edition III/2010, Feature Paper, 2010.
[15] G., Fettweis et al., “The tactile internet,” ITU-T Tech. Watch Rep., Tech. Rep., Aug. 2014.
[16] R. F., Schaefer, H., Boche, and H. V., Poor, “Secure communication under channel uncertainty and adversarial attacks,” Proc. IEEE, vol. 102, no. 10, pp. 1796–1813, Oct. 2015.
[17] D., Blackwell, L., Breiman, and A. J., Thomasian, “The capacity of a class of channels,” Ann. Math. Stat., vol. 30, no. 4, pp. 1229–1241, Dec. 1959.
[18] J., Wolfowitz, “Simultaneous channels,” Arch. Rational Mech. Analysis, vol. 4, no. 4, pp. 371–386, 1960.
[19] Y., Liang, G., Kramer, H. V., Poor, and S., Shamai (Shitz), “Compound wiretap channels,” EURASIP J. Wireless Commun. Netw., article ID 142374, pp. 1–13, 2009.
[20] 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.
[21] A., Khisti, “Interference alignment for the multiantenna compound wiretap channel,” IEEE Trans. Inf. Theory, vol. 57, no. 5, pp. 2976–2993, May 2011.
[22] E., Ekrem and S., Ulukus, “Degraded compound multi-receiver wiretap channels,” IEEE Trans. Inf. Theory, vol. 58, no. 9, pp. 5681–5698, Sep. 2012.
[23] 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.
[24] D., Blackwell, L., Breiman, and A. J., Thomasian, “The capacities of certain channel classes under random coding,” Ann. Math. Stat., vol. 31, no. 3, pp. 558–567, 1960.
[25] R., Ahlswede, “Elimination of correlation in random codes for arbitrarily varying channels,” Z. Wahrscheinlichkeitstheorie verw. Gebiete, vol. 44, pp. 159–175, 1978.
[26] I., Csiszár and P., Narayan, “The capacity of the arbitrarily varying channel revisited: Positivity, constraints,” IEEE Trans. Inf. Theory, vol. 34, no. 2, pp. 181–193, Mar. 1988.
[27] E., MolavianJazi, M., Bloch, and J. N., Laneman, “Arbitrary jamming can preclude secure communication,” in Proc. 47th Annual Allerton Conf. Commun., Control, Computing, Monticello, IL, USA, Sep. 2009, pp. 1069–1075.
[28] I., Bjelaković, H., Boche, and J., Sommerfeld, “Capacity results for arbitrarily varying wiretap channels,” Lecture Notes in Computer Science, vol. 7777, pp. 123–144, 2013.
[29] H., Boche and R. F., Schaefer, “Capacity results and super-activation for wiretap channels with active wiretappers,” IEEE Trans. Inf. Forensics Security, vol. 8, no. 9, pp. 1482–1496, Sep. 2013.
[30] M., Wiese, J., Nötzel, and H., Boche, “A channel under simultaneous jamming and eavesdropping attack – correlated random coding capacities under strong secrecy criteria,” IEEE Trans. Inf. Theory, vol. 62, no. 7, pp. 3844–3862, Jul. 2016.
[31] J., Nötzel, M., Wiese, and H., Boche, “The arbitrarily varying wiretap channel – secret randomness, stability and super-activation,” IEEE Trans. Inf. Theory, vol. 62, no. 6, pp. 3504–3531, Jun. 2016.
[32] H., Boche, R. F., Schaefer, and H. V., Poor, “On the continuity of the secrecy capacity of compound and arbitrarily varying wiretap channels,” IEEE Trans. Inf. Forensics Security, vol. 12, no. 10, pp. 2531–2546, Dec. 2015.
[33] Z., Goldfeld, P., Cuff, and H. H., Permuter, “Arbitrarily varying wiretap channels with type constrained states,” Jan. 2016. [Online]. Available:
[34] I., Csiszár, “Almost independence and secrecy capacity,” Probl. Pered. Inform., vol. 32, no. 1, pp. 48–57, 1996.
[35] U. M., Maurer and S., Wolf, “Information-theoretic key agreement: From weak to strong secrecy for free,” Lecture Notes in Computer Science, vol. 1807, pp. 351–368, 2000.
[36] R. F., Schaefer, H., Boche, and H. V., Poor, “Super-activation as a unique feature of arbitrarily varying wiretap channels,” in Proc. IEEE Int. Symp. Inf. Theory, Barcelona, Spain, Jul. 2016.
[37] I., Csiszár and P., Narayan, “Arbitrarily varying channels with constrained inputs and states,” IEEE Trans. Inf. Theory, vol. 34, no. 1, pp. 27–34, Jan. 1988.
[38] A., Ahlswede, I., Althöfer, C., Deppe, and U., Tamm, eds., Rudolf Ahlswede's Lectures on Information Theory 3 – Hiding Data: Selected Topics. New York: Springer, 2016.