# Secret Key Generation Among Multiple Terminals with Applications to Wireless Systems

Chapter

First Online:

## Abstract

The security of most existing cryptosystems relies on the (unproven) difficulty in solving a computational problem, e.g., factoring large integers or computing discrete logarithms in certain groups (cf. e.g.,[11]). This notion of security is called *computational complexity security*, as it is based on the assumption that an adversary has restricted computational power and lacks “efficient algorithms.„ However, this assumption is being weakened with the development of efficient algorithms as well as the increase in computational power of modern computers (e.g., quantum computer).

## Keywords

Channel State Information Steiner Tree LDPC Code Broadcast Channel Public Channel
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.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.R. Ahlswede and I. Csiszár, “Common randomness in information theory and cryptography, Part I: Secret sharing,”
*IEEE Trans. Inform. Theory*, vol. 39, pp. 1121–1132, July 1993.MATHCrossRefMathSciNetGoogle Scholar - 2.T. Aono, K. Higuchi, T. Ohira, B. Komiyama and H. Sasaoka, “Wireless secret key generation exploiting reactance-domain scalar response of multipath fading channels,”
*IEEE Trans. Antennas Propagation*, vol. 53, pp. 3776–3784, 2005.CrossRefGoogle Scholar - 3.J. Barros and M. R. D. Rodrigues, “Secrecy Capacity of Wireless Channels,”
*Proc. IEEE Int. Symp. Inform. Theory*, pp. 356–360, July 2006.Google Scholar - 4.C. H. Bennett, F. Bessette, G. Brassard, L. Salvail and J. Smolin, “Experimental quantum cryptography,”
*J. Cryptology*, vol. 5, pp. 3–28, 1992.MATHCrossRefGoogle Scholar - 5.C. H. Bennett, G. Brassard and J. M. Robert, “How to reduce your enemy’s information,”
*Advances in Cryptology - CRYPTO*, pp. 468–476, 1986.Google Scholar - 6.C. H. Bennett, G. Brassard and J. M. Robert, “Privacy amplification by public discussion,”
*SIAM J. Comput.*, vol. 17, pp. 210–229, Apr. 1988.CrossRefMathSciNetGoogle Scholar - 7.C. H. Bennett, G. Brassard, C. Crepeau and U. Maurer, “Generalized privacy amplification,”
*IEEE Trans. Inform. Theory*, vol. 41, pp. 1915–1923, Nov. 1995.MATHCrossRefMathSciNetGoogle Scholar - 8.M. Bloch, J. Barros, M. R. D. Rodrigues and S. W. McLaughlin, “Wireless informationtheoretic security–Part I: Theoretical aspects,” e-print arXiv: cs.IT/0611120, 2006.Google Scholar
- 9.M. Bloch, J. Barros, M. R. D. Rodrigues and S. W. McLaughlin, “Wireless informationtheoretic security–Part II: Practical implementation,” e-print arXiv: cs.IT/0611121, 2006.Google Scholar
- 10.G. Brassard and L. Salvail, “Secret-key reconciliation by public discussion,”
*Advances in Cryptology - EUROCRYPT*, pp. 410–423, 1994.Google Scholar - 11.J. A. Buchmann,
*Introduction to Cryptography*, New York: Springer, 2000.MATHGoogle Scholar - 12.C. Cachin and U. Maurer, “Linking information reconciliation and privacy amplification,”
*J. Cryptology*, vol. 10, pp. 97–110, 1997.MATHCrossRefGoogle Scholar - 13.J. L. Carter and M. N. Wegman, “Universal classes of hash functions,”
*J. Comput. Syst. Scien.*, vol. 18, pp. 143–154, 1979.MATHCrossRefMathSciNetGoogle Scholar - 14.Y. Chen and A. J. Han Vinck, “Wiretap channel with side information,”,
*Proc. Int. Symp. Inform. Theory*, pp. 2607–2611, July 2006.Google Scholar - 15.J. Chen, D. He and E. Yang, “On the codebook-level duality between Slepian-Wolf coding and channel coding,”
*Proc. IEEE Inform. Theory Appl. Workshop*, pp. 84–93, Feb. 2007.Google Scholar - 16.T. P. Coleman, A. H. Lee, M. M´edard, and M. Effros, “Low-Complexity Approaches to Slepian-Wolf Near-Lossless Distributed Data Compression,”
*IEEE Trans. Inform. Theory*, vol. 52, pp. 3546–3561, Aug. 2006.CrossRefMathSciNetGoogle Scholar - 17.R. Cramer, Y. Dodis, S. Fehr, C. Padr´o and D. Wichs, “Detection of algebraic manipulation with applications to robust secret sharing and fuzzy extractors,”
*Advances in Cryptology - EUROCRYPT*, Apr. 2008.Google Scholar - 18.I. Csiszár and J. Körner, “Broadcast channels with confidential messages,”
*IEEE Trans. Inform. Theory*, vol. IT-24, pp. 339–348, May 1978.CrossRefGoogle Scholar - 19.I. Csiszár and J. Körner,
*Information Theory: Coding Theorems for Discrete Memoryless Systems*. Academic, New York, N.Y., 1982.Google Scholar - 20.I. Csiszár and P. Narayan, “Common randomness and secret key generation with a helper,”
*IEEE Trans. Inform. Theory*, vol. 46, pp. 344–366, Mar. 2000.MATHCrossRefMathSciNetGoogle Scholar - 21.I. Csiszár and P. Narayan, “Secrecy capacities for multiple terminals,”
*IEEE Trans. Inform. Theory*, vol. 50, pp. 3047–3061, Dec. 2004.CrossRefMathSciNetGoogle Scholar - 22.I. Csiszár and P. Narayan, “Secrecy capacities for multiterminal channel models,”
*IEEE Trans. Inform. Theory*, Jun. 2008.Google Scholar - 23.Y. Dodis, J. Katz, L. Reyzin and A. Smith, “Robust fuzzy extractors and authenticated key agreement from close secrets,”
*Advances in Cryptology - CRYTPO*, Aug. 2006.Google Scholar - 24.Y. Dodis, R. Ostrovsky, L. Reyzin and A. Smith, “Fuzzy extractors: How to generate strong keys from biometrics and other noisy data,”
*SIAM J. Comput.*, pp. 97–139, 2008.Google Scholar - 25.H. N. Gabow and H. H. Westermann, “Forests, frames, and games: algorithms for matroid sums and applications,”
*Algorithmica*, 7: pp. 465–497, 1992.MATHCrossRefMathSciNetGoogle Scholar - 26.P. Gács and J. Körner, “Common information is far less than mutual information,”
*Probl. Contr. Inform. Theory*, vol. 2, pp. 149–162, 1973.MATHGoogle Scholar - 27.J. Garcia-Frias and Y Zhao, “Compression of correlated binary sources using turbo codes,”
*IEEE Commun. Lett.*, vol. 5, pp. 417–419, Oct. 2001.CrossRefGoogle Scholar - 28.A. A. Gohari and V. Anantharam, “Information-theoretic key agreement of multiple terminals—Part I: Source model,”
*IEEE Trans. Inform. Theory*, submitted.Google Scholar - 29.A. A. Gohari and V. Anantharam, “Information-theoretic key agreement of multiple terminals—Part II: Channel model,”
*IEEE Trans. Inform. Theory*, submitted.Google Scholar - 30.P. Gopala, L. Lai and H. El Gamal, “On the Secrecy Capacity of Fading Channels,” e-print arXiv: cs.IT/0610103, 2006.Google Scholar
- 31.J. Grubb, S. Vishwanath, Y. Liang and H. V. Poor, “Secrecy capacity for semideterministic wire-tap channels,”
*Proc. IEEE Inform. Theory Workshop Wireless Networks*, 2007.Google Scholar - 32.A. A. Hassan, W. E. Stark, J. E. Hershey and S. Chennakeshu, “Cryptographic key agreement for mobile radio,”
*IEEE Digital Signal Processing Mag.*, vol. 6, pp. 207-212, 1996.CrossRefGoogle Scholar - 33.J. E. Hershey, A. A. Hassan and R. Yarlagadda, “Unconventional cryptographic keying variable management,”
*IEEE Trans. Commun.*, vol. 43, pp. 3–6, Jan. 1995.MATHCrossRefGoogle Scholar - 34.H. Imai, K. Kobara and K. Morozov, “On the possibility of key agreement using variable directional antenna,”
*Proc. Joint Workshop Inform. Security*, 2006.Google Scholar - 35.A. Khisti, A. Tchamkerten and G. W. Wornell, “Secure broadcasting,” e-print arXiv: cs.IT/0702093, 2007.Google Scholar
- 36.A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas: The MISOME wiretap channel,” e-print arXiv: cs.IT/07084219, 2007.Google Scholar
- 37.A. Khisti, G. W. Wornell, A. Wiesel and Y. Eldar, “On the Gaussian MIMI wiretap channel,”
*Proc. IEEE Int. Symp. Inform. Theory*, pp. 2471–2475, Jun. 2007.Google Scholar - 38.H. Kooraparty, A. A. Hassan and S. Chennakeshu, “Secure information transmission for mobile radio,”
*IEEE Commun. Lett.*, vol. 4, pp. 52–55, Feb. 2000.CrossRefGoogle Scholar - 39.L. Lai and H. El Gamal, “The relay-eavesdropper channel: Cooperation for secrecy,”
*IEEE Trans. Inform. Theory*, submitted.Google Scholar - 40.L. Lai, H. El Gamal and H. V. Poor, “The wiretap channel with feedback: Encryption over the channel,” e-print arXiv: cs.IT/07042259, 2007.Google Scholar
- 41.S. L. Leung-Yan-Cheong and M. Hellman, “The Gaussian wire-tap channel,”
*IEEE Trans. Inform. Theory*, vol. 24, pp. 451–456, July 1978.MATHCrossRefMathSciNetGoogle Scholar - 42.Z. Li, R. Yates and W. Trappe, “Secrecy capacity of independent parallel channels,”
*Proc. Allerton Conf. Commun. Control, Comput.*, Sept. 2006.Google Scholar - 43.Z. Li, R. Yates and W. Trappe, “Secure communication with a fading eavesdropper channel,”
*Proc. IEEE Int. Symp. Inform. Theory*, pp. 1296–1300, Jun. 2007.Google Scholar - 44.Z. Li, W. Trappe and R. Yates, “Secret communication via multi-antenna transmission,”
*Proc. Conf. Inform. Scien. Syst.*, Mar. 2007.Google Scholar - 45.Y. Liang and H. V. Poor, “Multiple access channels with confidential messages,”
*IEEE Trans. Inform. Theory*, vol. 54, pp. 976–1002, Mar. 2008.CrossRefMathSciNetGoogle Scholar - 46.Y. Liang, H. V. Poor and S. Shamai, “Secure communication over fading channels,”
*IEEE Trans. Inform. Theory*, Jun. 2008.Google Scholar - 47.R. Liu, Y. Liang, H. V. Poor and P. Spasojevic, “Secure nested codes for Type II wiretap channels,”
*Proc. IEEE Inform. Theory Workshop*, pp. 337–342, Sept. 2007.Google Scholar - 48.R. Liu, I. Marić, R. Yates and P. Spasojević, “The discrete memoryless multiple access channel with confidential messages,”
*Proc. Int. Symp. on Inform. Theory*, pp. 957–961, July 2006.Google Scholar - 49.R. Liu, I. Marić, P. Spasojević and R. Yates, “Discrete memoryless interference and broadcast channels with confidential messages: Secrecy capacity regions,”
*IEEE Trans. Inform. Theory*, Jun. 2008.Google Scholar - 50.R. Liu and H. V. Poor, “Secrecy capacity region of a multi-antenna Gaussian broadcast channel with confidential messages,” e-print arXiv: cs.IT/07094671, 2007.Google Scholar
- 51.A. D. Liveris, Z. Xiong, C. N. Georghiades, “Compression of binary sources with side information at the decoding using LDPC codes,”
*IEEE Commun. Lett.*, vol. 6, pp. 440–442, Oct. 2002.CrossRefGoogle Scholar - 52.S. Mathur, W. Trappe, N. Mandayam, C. Ye and A. Reznik, “Radio-telepathy: Extracting a secret key from an unauthenticated wireless channel,”
*Proc. ACM Conf. Mobile Comput. Network.*, Sept. 2008.Google Scholar - 53.U. Maurer, “Secret key agreement by public discussion from common information,”
*IEEE Trans. Inform. Theory*, vol. 39, pp. 733–742, May 1993.MATHCrossRefMathSciNetGoogle Scholar - 54.U. M. Maurer, “The strong secret key rate of discrete random triples,”
*Communications and Cryptography: Two Sides of One Tapestry*, R. E. Blahut et al., Ed., Kluwer, Norwell, MA, Ch. 26, pp. 271–285, 1994.Google Scholar - 55.U. M. Maurer, “Information-theoretically secure secret-key agreement by NOT authenticated public discussion,” in
*Advances in Cryptology - EUROCRYPT*, 1997.Google Scholar - 56.U. M. Maurer and S. Wolf, “Information-theoretic key agreement: from weak to strong secrecy for free,”
*Advances in Cryptology - EUROCRYPT*, pp. 351–368, May 2000.Google Scholar - 57.U. Maurer and S. Wolf, “Secret-key agreement over unauthenticated public channels— Part I: Definitions and a completeness result,”
*IEEE Trans. Inform. Theory*, vol. 49, pp. 822–831, Apr. 2003.MATHCrossRefMathSciNetGoogle Scholar - 58.U. Maurer and S. Wolf, “Secret-key agreement over unauthenticated public channels— Part II: The simulatability condition,”
*IEEE Trans. Inform. Theory*, vol. 49, pp. 832–838, Apr. 2003.MATHCrossRefMathSciNetGoogle Scholar - 59.U. Maurer and S. Wolf, “Secret-key agreement over unauthenticated public channels— Part III: Privacy amplification,”
*IEEE Trans. Inform. Theory*, vol. 49, pp. 839–851, Apr. 2003.MATHCrossRefMathSciNetGoogle Scholar - 60.C. Mitrpant, A. J. H. Vinck and Y. Luo, “An achievable region for the Gaussian wiretap channel with side information,”
*IEEE Trans. Inform. Theory*, vol. 52, pp. 2181–2190, May 2006.CrossRefMathSciNetGoogle Scholar - 61.J. Muramatsu, “Secret key agreement from correlated source outputs using LDPC matrices,”
*IEICE Trans. Fundamentals*, vol. E89-A, pp. 2036–2046, July 2006.CrossRefGoogle Scholar - 62.C. St. J. A. Nash-Williams, “Edge disjoint spanning trees of finite graphs,”
*J. London Math. Soc.*, 36, pp. 445–450, 1961.MATHCrossRefMathSciNetGoogle Scholar - 63.S. Nitinawarat, C. Ye, A. Barg, P. Narayan and A. Reznik, “Secret key generation for a pairwise independent network model,”
*Proc. Int. Symp. Inform. Theory*, pp. 1015–1019, July 2008.Google Scholar - 64.P. Parada and R. Blahut, “Secrecy capacity of SIMO and slow fading channels,”
*Proc. IEEE Int. Symp. Inform. Theory*, pp. 2152–2155, Sept. 2005.Google Scholar - 65.S. S. Pradhan and K. Ramchandran, “Distributed source coding using syndromes (DISCUS): Design and construction,”
*IEEE Trans. Inform. Theory*, vol. 49, pp. 626–643, Mar. 2003.MATHCrossRefMathSciNetGoogle Scholar - 66.R. Raz, I. Reingold and S. Vadhan, “Extracting all the randomness and reducing the error in Trevisan’s extractors,”
*Proc. Symp. Theory of Comput.*, pp. 149–158, 1999.Google Scholar - 67.R. Renner and S. Wolf, “New bounds in secret-key agreement: the gap between formation and secrecy extraction,”
*Advances in Cryptology - EUROCRYPT*, pp. 562–577, 2003.Google Scholar - 68.A. Reznik, A. Carlton, A. Briancon, Y. Shah, P. Chitrapu, R. Mukherjee and M. Rudolf, “Method and system for securing wireless communications,” U.S. patent application 20060133338, 11/283017, Jun. 2006.Google Scholar
- 69.A. Schrijver,
*Theory of Linear and Integer Programming*, New York: John Wiley and Sons, 1986.MATHGoogle Scholar - 70.A. Schrijver,
*Combinatorial Optimization — Polyhedra and Efficiency*, New York: Springer, 2003.MATHGoogle Scholar - 71.S. Shafiee and S. Ulukus, “Achievable rates in Gaussian MISO channels with secrecy constraints,”
*Proc. IEEE Int. Symp. Inform. Theory*, pp. 2466–2470, June. 2007.Google Scholar - 72.C. E. Shannon, “Communication theory of secrecy systems,”
*Bell Syst. Tech. J.*, vol. 28, pp. 656–715, Oct. 1949.MATHMathSciNetGoogle Scholar - 73.X. Tang, R. Liu, P. Spasojevic and H. V. Poor, “Interference-assisted secret communication,”,
*Proc. IEEE Inform. Theory Workshop*, May 2008.Google Scholar - 74.E. Tekin and A. Yener, “The Gaussian multiple access wire-tap channel with collective secrecy constraints,”
*Proc. Int. Symp. Inform. Theory*, pp. 1164–1168, July 2006.Google Scholar - 75.A. Thangaraj, S. Dihidar, A. R. Calderbank, S. McLaughlin and J. M. Merolla, “Capacity achieving codes for the wiretap channel with applications to quantum key distribution,” e-print arXiv: cs.IT/0411003, 2004.Google Scholar
- 76.W. T. Tutte, “On the problem of decomposing a graph into
*n*connected factors,”*J. London Math. Soc.*, 36, pp. 221–230, 1961.MATHCrossRefMathSciNetGoogle Scholar - 77.M. N. Wegman and J. Carter, “New hash functions and their use in authentication and set equality,”
*J. Comput. Syst. Scien.*, vol. 22, pp. 265–279, 1981.MATHCrossRefMathSciNetGoogle Scholar - 78.R. Wilson, D. Tse and R. Scholtz, “Channel identification: Secret sharing using reciprocity in ultrawideband channels,”
*IEEE Trans. Inform. Foren. and Security*, vol. 2, pp. 364–375, Sept. 2007.CrossRefGoogle Scholar - 79.A. D. Wyner, “The wire-tap channel,”
*Bell Syst. Tech. J.*, vol. 54, pp. 1355–1387, Oct. 1975.MathSciNetGoogle Scholar - 80.L. Xiao, L. Greenstein, N. Mandayam and W. Trappe, “Using the physical layer for wireless authentication under time-variant channels,”
*IEEE Trans. Wireless Commun.*, to appear.Google Scholar - 81.C. Ye and P. Narayan, “The private key capacity region for three terminals,”
*Proc. Int. Symp. on Inform. Theory*, p 44, Jun. 2004.Google Scholar - 82.C. Ye and P. Narayan, “Secret key and private key constructions for simple multiterminal source models,”
*Proc. Int. Symp. Inform. Theory*, pp. 2133–2137, Sept. 2005.Google Scholar - 83.C. Ye and P. Narayan, “The secret key-private key capacity region for three terminals,”
*Proc. IEEE Int. Symp. Inform. Theory*, pp. 2142–2146, Sept. 2005.Google Scholar - 84.C. Ye, A. Reznik and Y. Shah, “Extracting secrecy from jointly Gaussian random variables,”
*Proc. Int. Symp. Inform. Theory*, pp. 2593–2597, July 2006.Google Scholar - 85.C. Ye and A. Reznik, “Group secret key generation algorithms,”
*Proc. Int. Symp. Inform. Theory*, pp. 2596–2600, Jun. 2007.Google Scholar - 86.C. Ye, A. Reznik, Y. Shah and G. Sternberg, “Method and system for generating a secret key from joint randomness,” U.S. patent application 20070165845, 11/612671, July 2007.Google Scholar
- 87.M. Yuksel and E. Erkip, “The relay channel with a wire-tapper,”
*Proc. Conf. Inform. Scien. Syst.*, Mar. 2007.Google Scholar

## Copyright information

© Springer Science+Business Media, LLC 2009