Abstract
Cayley hash functions are a family of cryptographic hash functions constructed from the Cayley graphs of non-Abelian finite groups. Their security relies on the hardness of mathematical problems related to long-standing conjectures in graph and group theory. We recall the Cayley hash design and known results on the underlying problems. We then describe related open problems, including the cryptanalysis of relevant parameters as well as new applications to cryptography and outside, assuming either that the problem is “hard” or easy.
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References
Babai, L., Kantor, W.M., Lubotzky, A.: Small-diameter Cayley graphs for finite simple groups. European J. Combin. 10, 507–552 (1989)
Babai, L., Seress, Á.: On the diameter of permutation groups. European J. Combin. 13(4), 231–243 (1992)
Babai, L., Hayes, T.P.: Near-independence of permutations and an almost sure polynomial bound on the diameter of the symmetric group. In: SODA, pp. 1057–1066. SIAM (2005)
Babai, L., Hetyei, G., Kantor, W.M., Lubotzky, A., Seress, Á.: On the diameter of finite groups. In: FOCS, vol. II, pp. 857–865. IEEE (1990)
Bourgain, J., Gamburd, A.: Uniform expansion bounds for cayley graphs of \(sl_2(\mathbb{F}_p)\). Ann. Math. 167(2), 625–642 (2008)
Breuillard, E., Green, B., Tao, T.,Approximate subgroups of linear groups. arXiv:1005.1881v1, May 2010
Charles, D.X., Lauter, K.E., Goren, E.Z.: Cryptographic hash functions from expander graphs. J. Cryptology 22(1), 93–113 (2009)
Damgård, I.B.: Collision free hash functions and public key signature schemes. In: Price, W.L., Chaum, D. (eds.) EUROCRYPT 1987. LNCS, vol. 304, pp. 203–216. Springer, Heidelberg (1988)
de Meulenaer, G., Petit, C., Quisquater, J.-J.: Hardware implementations of a variant of the Zmor-Tillich hash function: can a provably secure hash function be very efficient ? Cryptology ePrint Archive, Report /229 (2009). http://eprint.iacr.org/
Dinai, O.: Poly-log diameter bounds for some families of finite groups. Proc. Amer. Math. Soc. 134, 3137–3142 (2006)
Even, S., Goldreich, O.: The minimum-length generator sequence problem is NP-hard. J. Algorithms 2(3), 311–313 (1981)
Goldwasser, S., Micali, S., Rivest, R.L.: A “paradoxical” solution to the signature problem (extended abstract). In: FOCS, pp. 441–448. IEEE (1984)
Grassl, M., Ilic, I., Magliveras, S.S., Steinwandt, R.: Cryptanalysis of the Tillich-Zémor hash function. J. Cryptology 24(1), 148–156 (2011)
Helfgott, H., Seress, A.: On the diameter of permutation groups (2011). http://arxiv.org/abs/1109.3550
Helfgott, H.A.: Growth, generation in \(SL_2(Z, pZ)\). Ann. Math. 167(2), 601–623 (2008)
Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Amer. Math. Soc. 43, 439–561 (2006)
Jerrum, M.R.: The complexity of finding minimum-length generator sequences. Theor. Comput. Sci. 36(2–3), 265–289 (1985)
Kantor, W.M.: Some large trivalent graphs having small diameters. Discrete Appl. Math. 37(38), 353–357 (1992)
Kassabov, M., Riley, T.R.: Diameters of Cayley graphs of Chevalley groups. Eur. J. Comb. 28(3), 791–800 (2007)
Landau, Z., Russell, A.: Random cayley graphs are expanders: a simple proof of the alon-roichman theorem. Electr. J. Comb. 11(1) (2004)
Larsen, M.: Navigating the Cayley graph of \(SL_2(\mathbb{F}_p)\). Int. Math. Res. Not. IMRN 27, 1465–1471 (2003)
Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8, 261–277 (1988)
Lubotzky, A.: Expander graphs in pure and applied mathematics. Bull. Amer. Math. Soc. 49, 113–162 (2012)
Petit, C.: Towards factoring in \(SL(2,\mathbb{F}_{2^n})\). Design, Codes and Cryptography, September 2012. doi: 10.1007/s10623-012-9743-x
Petit, C., Lauter, K., Quisquater, J.-J.: Full cryptanalysis of LPS and Morgenstern hash functions. In: Ostrovsky, R., De Prisco, R., Visconti, I. (eds.) SCN 2008. LNCS, vol. 5229, pp. 263–277. Springer, Heidelberg (2008)
Petit, C., Lauter, K.E., Quisquater, J.-J.: Cayley hashes: a class of efficient graph-based hash functions (2007). http://perso.uclouvain.be/christophe.petit/index.html
Petit, C., Quisquater, J.-J.: Preimages for the Tillich-Zémor hash function. In: Biryukov, A., Gong, G., Stinson, D.R. (eds.) SAC 2010. LNCS, vol. 6544, pp. 282–301. Springer, Heidelberg (2011)
Petit, C., Quisquater, J.-J.: Rubik’s for cryptographers. Not. Am. Math. Soc. 60, 733–739 (2013)
Pyber, L., Szab, E.: Growth in finite simple groups of Lie type. arXiv:1001.4556v1, January 2010
Riley, T.R.: Navigating in the Cayley graphs of \(SL_N(\mathbb{Z})\) and \(SL_N(\mathbb{F}_p)\). Geom. Dedicata 113(1), 215–229 (2005)
Mullan, C., Blackburn, S.R., Cid, C.: Group theory in cryptography (2010). http://arxiv.org/abs/0906.5545
Steinwandt, R., Grassl, M., Geiselmann, W., Beth, T.: Weaknesses in the \(SL_2(\mathbb{F}_{2^{n}})\) hashing scheme. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 287–299. Springer, Heidelberg (2000)
Tillich, J.-P., Zémor, G.: Group-theoretic hash functions. In: Cohen, G., Lobstein, A., Zémor, G., Litsyn, S.N. (eds.) Algebraic Coding 1993. LNCS, vol. 781, pp. 90–110. Springer, London (1994)
Tillich, J.-P., Zémor, G.: Hashing with \(SL_2\). In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 40–49. Springer, Heidelberg (1994)
Tillich, J.-P., Zémor, G.: Collisions for the LPS expander graph hash function. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 254–269. Springer, Heidelberg (2008)
Wiener, M.J.: Cryptanalysis of short RSA secret exponents. IEEE Trans. Inf. Theory 36(3), 553–558 (1990)
Zémor, G.: Hash functions and graphs with large Girths. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 508–511. Springer, Heidelberg (1991)
Acknowledgements
Part of this work was done while Christophe Petit was visiting the Computer Science Department at University College London and the Number Theory Group at the University of Oxford, under an FRS-FNRS Research Collaborator grant at Universit catholique de Louvain. He is grateful to Jens Groth (UCL) and Alan Lauder (Oxford) for the fruitful work he could do there. The research leading to these results has also received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 307937 and the Engineering and Physical Sciences Research Council grant EP/J009520/1.
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Petit, C., Quisquater, JJ. (2016). Cryptographic Hash Functions and Expander Graphs: The End of the Story?. In: Ryan, P., Naccache, D., Quisquater, JJ. (eds) The New Codebreakers. Lecture Notes in Computer Science(), vol 9100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49301-4_19
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