Abstract
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.
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Notes
- 1.
In this paper, the expression “Eve does not know key k” means the key is completely unknown to Eve; that is, she cannot determine whether \(k=0\) or \(k=1\) with a probability of more than 1/2.
- 2.
Note that the notation \(\mathcal {P}\) in this paper denotes a protocol, not a power set.
References
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). https://doi.org/10.1007/978-3-319-07536-5_8
Bennett, C.H., Brassard, G., Crépeau, C., Maurer, U.M.: Generalized privacy amplification. IEEE Trans. Inf. Theory 41(6), 1915–1923 (1995). https://doi.org/10.1109/18.476316
Colbourn, C.J., Colbourn, C.: The Combinatorics of Network Reliability, vol. 200. Oxford University Press, New York (1987)
Csiszár, I., Narayan, P.: Secrecy capacities for multiple terminals. IEEE Trans. Inf. Theory 50(12), 3047–3061 (2004). https://doi.org/10.1109/TIT.2004.838380
Dolev, D., Dwork, C., Waarts, O., Yung, M.: Perfectly secure message transmission. J. ACM 40(1), 17–47 (1993). https://doi.org/10.1145/138027.138036
Franklin, M.K., Wright, R.N.: Secure communication in minimal connectivity models. J. Cryptol. 13(1), 9–30 (2000). https://doi.org/10.1007/s001459910002
Franklin, M.K., Yung, M.: Secure hypergraphs: Privacy from partial broadcast. SIAM J. Discrete Math. 18(3), 437–450 (2004). https://doi.org/10.1137/S0895480198335215
Indo, Y., Mizuki, T., Nishizeki, T.: Absolutely secure message transmission using a key sharing graph. Discrete Math. Alg. Appl. 4(4) (2012). https://doi.org/10.1142/S179383091250053X
Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Theory of Graphs: International Symposium, pp. 215–232 (1967)
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). https://doi.org/10.1142/S0129054111008659
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). https://doi.org/10.1016/j.ipl.2009.04.004
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). https://doi.org/10.1007/978-3-642-17773-6_7
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). https://doi.org/10.1007/978-3-319-12060-7_7
Vernam, G.S.: Cipher printing telegraph systems for secret wire and radio telegraphic communications. Trans. Am. Inst. Electr. Eng. XLV, 295–301 (1926)
Wang, Y., Desmedt, Y.: Secure communication in multicast channels: The answer to franklin and wright’s question. J. Cryptol. 14(2), 121–135 (2001). https://doi.org/10.1007/s00145-001-0002-y
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). http://search.ieice.org/bin/summary.php?id=e93-a_9_1650
Acknowledgement
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.
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Sasaki, T., Agbor, B.M., Masuda, S., Hayashi, Yi., Mizuki, T., Sone, H. (2018). Secret Key Amplification from Uniformly Leaked Key Exchange Complete Graph. In: Rahman, M., Sung, WK., Uehara, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2018. Lecture Notes in Computer Science(), vol 10755. Springer, Cham. https://doi.org/10.1007/978-3-319-75172-6_3
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