Abstract
The factorization problem in non-abelian groups is still an open and a difficult problem [12]. The hardness of the problem is illustrated by the moves of the Rubik’s cube. We will define a public key identification scheme based on this problem, in the case of the Rubik’s cube, when the number of moves is fixed to a given value. Our scheme consists of an interactive protocol which is zero-knowledge argument of knowledge under the assumption of the existence of a commitment scheme. We will see that our scheme works with any non-abelian groups with a set of authorized moves that has a specific property. Then we will generalize the scheme for larger Rubik’s cubes and for any groups.
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References
Colmez, P.: Le Rubik’s cube, groupe de poche. ENS Ulm (May 2010)
Demaine, E.D., Demaine, M.L., Eisenstat, S., Lubiw, A., Winslow, A.: Algorithms for Solving Rubik’s Cubes. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 689–700. Springer, Heidelberg (2011)
Fiat, A., Shamir, A.: How to prove yourself: Practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987)
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completness, 2nd edn. W.H Freeman and Co. (1991, 1979)
Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. J. ACM 38, 690–728 (1991)
Goldreich, O., Oren, Y.: Definitions and properties of Zero-knowledge proof systems. Journal of Cryptology 7(1), 1–32 (1994)
Halevi, S., Micali, S.: Practical and Provably-Secure Commitment Schemes from Collision-Free Hashing. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 201–215. Springer, Heidelberg (1996)
Joyner, D.: Adventures with Group Theory: Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys, 2nd edn. The Johns Hopkins University Press (2008)
Kendall, G., Parkes, A.J., Spoerer, K.: A Survey of NP-Complete Puzzles. ICGA Journal 31(1), 13–34 (2008)
Kozen, D.: Lower bounds for natural proof systems. In: FOCS, pp. 254–266 (1977)
Lang, S.: Algebra Revised. 3rd edn. Addison-Wesley (2002)
Petit, C., Quisquater, J.-J.: Rubik’s for cryptographers. IACR Cryptology ePrint Archive, 2011:638 (2011)
Pointcheval, D.: A New Identification Scheme based on the Perceptrons Problem. In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 319–328. Springer, Heidelberg (1995)
Pyber, L., Szabó, E.: Growth in finite simple groups of Lie type of bounded rank. ArXiv e-prints (May 2010)
Rokicki, T., Kociemba, H., Davidson, M., Dethrige, J.: God’s number is 20, http://cube20.org
Stein, W.A., et al.: Sage Mathematics Software (Version 4.7-OSX-32bit-10.5). The Sage Development Team (2011), http://www.sagemath.org
Stern, J.: A New Identification Scheme Based on Syndrome Decoding. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 13–21. Springer, Heidelberg (1994)
Stern, J.: Designing Identification Schemes with Keys of Short Size. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 164–173. Springer, Heidelberg (1994)
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Volte, E., Patarin, J., Nachef, V. (2013). Zero Knowledge with Rubik’s Cubes and Non-abelian Groups. In: Abdalla, M., Nita-Rotaru, C., Dahab, R. (eds) Cryptology and Network Security. CANS 2013. Lecture Notes in Computer Science, vol 8257. Springer, Cham. https://doi.org/10.1007/978-3-319-02937-5_5
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DOI: https://doi.org/10.1007/978-3-319-02937-5_5
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