Secret Key Amplification from Uniformly Leaked Key Exchange Complete Graph

  • Tatsuya SasakiEmail author
  • Bateh Mathias Agbor
  • Shingo Masuda
  • Yu-ichi Hayashi
  • Takaaki Mizuki
  • Hideaki Sone
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10755)


We assume that every pair of n players has shared a one-bit key in advance, and that each key has been completely exposed to an eavesdropper, Eve, independently with a fixed probability p (and, thus, is perfectly secure with a probability of \(1-p\)). Using these pre-shared, possibly leaked keys, we want two designated players to share a common one-bit secret key in cooperation with other players so that Eve’s knowledge about the generated secret key will be as small as possible. The existing protocol, called the st-flow protocol, achieves this, but the specific probability that Eve knows the generated secret key is unknown. In this study, we answer this problem by showing the exact leak probability as a polynomial in p for any number n of players.


Key exchange graph st-numbering Key agreement protocol Privacy amplification Network reliability problem 



We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper. We thank Mr. Shigehiro Matsuda for his valuable discussions. This work was supported by JSPS KAKENHI Grant Number 15K11983.


  1. 1.
    Ahmadi, H., Safavi-Naini, R.: Private message transmission using disjoint paths. In: Boureanu, I., Owesarski, P., Vaudenay, S. (eds.) ACNS 2014. LNCS, vol. 8479, pp. 116–133. Springer, Cham (2014). Google Scholar
  2. 2.
    Bennett, C.H., Brassard, G., Crépeau, C., Maurer, U.M.: Generalized privacy amplification. IEEE Trans. Inf. Theory 41(6), 1915–1923 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Colbourn, C.J., Colbourn, C.: The Combinatorics of Network Reliability, vol. 200. Oxford University Press, New York (1987)zbMATHGoogle Scholar
  4. 4.
    Csiszár, I., Narayan, P.: Secrecy capacities for multiple terminals. IEEE Trans. Inf. Theory 50(12), 3047–3061 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dolev, D., Dwork, C., Waarts, O., Yung, M.: Perfectly secure message transmission. J. ACM 40(1), 17–47 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Franklin, M.K., Wright, R.N.: Secure communication in minimal connectivity models. J. Cryptol. 13(1), 9–30 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Franklin, M.K., Yung, M.: Secure hypergraphs: Privacy from partial broadcast. SIAM J. Discrete Math. 18(3), 437–450 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Indo, Y., Mizuki, T., Nishizeki, T.: Absolutely secure message transmission using a key sharing graph. Discrete Math. Alg. Appl. 4(4) (2012).
  9. 9.
    Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Theory of Graphs: International Symposium, pp. 215–232 (1967)Google Scholar
  10. 10.
    Mizuki, T., Nakayama, S., Sone, H.: An application of st-numbering to secret key agreement. Int. J. Found. Comput. Sci. 22(5), 1211–1227 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mizuki, T., Sato, T., Sone, H.: A one-round secure message broadcasting protocol through a key sharing tree. Inf. Process. Lett. 109(15), 842–845 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nagaraja, S.: Privacy amplification with social networks. In: Christianson, B., Crispo, B., Malcolm, J.A., Roe, M. (eds.) Security Protocols 2007. LNCS, vol. 5964, pp. 58–73. Springer, Heidelberg (2010). CrossRefGoogle Scholar
  13. 13.
    Ošt’ádal, R., Švenda, P., Matyáš, V.: A new approach to secrecy amplification in partially compromised networks (invited paper). In: Chakraborty, R.S., Matyas, V., Schaumont, P. (eds.) SPACE 2014. LNCS, vol. 8804, pp. 92–109. Springer, Cham (2014). Google Scholar
  14. 14.
    Vernam, G.S.: Cipher printing telegraph systems for secret wire and radio telegraphic communications. Trans. Am. Inst. Electr. Eng. XLV, 295–301 (1926)CrossRefGoogle Scholar
  15. 15.
    Wang, Y., Desmedt, Y.: Secure communication in multicast channels: The answer to franklin and wright’s question. J. Cryptol. 14(2), 121–135 (2001). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Watanabe, S., Matsumoto, R., Uyematsu, T.: Strongly secure privacy amplification cannot be obtained by encoder of slepian-wolf code. IEICE Trans. 93(9), 1650–1659 (2010). CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Tatsuya Sasaki
    • 1
    Email author
  • Bateh Mathias Agbor
    • 1
  • Shingo Masuda
    • 1
  • Yu-ichi Hayashi
    • 2
  • Takaaki Mizuki
    • 3
  • Hideaki Sone
    • 3
  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.Nara Institute of Science and TechnologyIkoma, NaraJapan
  3. 3.Cyberscience CenterTohoku UniversitySendaiJapan

Personalised recommendations