Abstract
The Rainbow Signature Scheme is a non-trivial generalization of the well known Unbalanced Oil and Vinegar Signature Scheme (Eurocrypt ’99) minimizing the length of the signatures. Recently a new variant based on non-commutative rings, called NC-Rainbow, was introduced at CT-RSA 2012 to further minimize the secret key size. We disprove the claim that NC-Rainbow is as secure as Rainbow in general and show how to reduce the complexity of MinRank attacks from 2288 to 2192 and of HighRank attacks from 2128 to 296 for the proposed instantiation over the ring of Quaternions. We further reveal some facts about Quaternions that increase the complexity of the signing algorithm. We show that NC-Rainbow is just a special case of introducing further structure to the secret key in order to decrease the key size. As the results are comparable with the ones achieved by equivalent keys, which provably do not decrease security, and far worse than just using a PRNG, we recommend not to use NC-Rainbow.
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References
Aristidou, M., Demetre, A.: A Note on Quaternion Rings over ℤ p . International Journal of Algebra 3, 725–728 (2009)
Billet, O., Gilbert, H.: Cryptanalysis of Rainbow. In: De Prisco, R., Yung, M. (eds.) SCN 2006. LNCS, vol. 4116, pp. 336–347. Springer, Heidelberg (2006)
Computational Algebra Group, University of Sydney. The MAGMA Computational Algebra System for Algebra, Number Theory and Geometry, http://magma.maths.usyd.edu.au/magma/
Ding, J., Schmidt, D.: Rainbow, a New Multivariable Polynomial Signature Scheme. In: Ioannidis, J., Keromytis, A.D., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005)
Ding, J., Yang, B.-Y., Chen, C.-H.O., Chen, M.-S., Cheng, C.-M.: New Differential-Algebraic Attacks and Reparametrization of Rainbow. In: Bellovin, S.M., Gennaro, R., Keromytis, A.D., Yung, M. (eds.) ACNS 2008. LNCS, vol. 5037, pp. 242–257. Springer, Heidelberg (2008)
Faugère, J.-C., Levy-dit-Vehel, F., Perret, L.: Cryptanalysis of MinRank. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 280–296. Springer, Heidelberg (2008)
Goubin, L., Courtois, N.T.: Cryptanalysis of the TTM Cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000)
Kipnis, A., Patarin, J., Goubin, L.: Unbalanced Oil and Vinegar Signature Schemes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 206–222. Springer, Heidelberg (1999)
Kipnis, A., Shamir, A.: Cryptanalysis of the Oil & Vinegar Signature Scheme. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 257–266. Springer, Heidelberg (1998)
Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. EMA, vol. 20. CUP, Cambridge (1997)
Petzoldt, A., Bulygin, S., Buchmann, J.: CyclicRainbow – A Multivariate Signature Scheme with a Partially Cyclic Public Key. In: Gong, G., Gupta, K.C. (eds.) INDOCRYPT 2010. LNCS, vol. 6498, pp. 33–48. Springer, Heidelberg (2010)
Petzoldt, A., Bulygin, S., Buchmann, J.: Selecting Parameters for the Rainbow Signature Scheme. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 218–240. Springer, Heidelberg (2010)
Petzoldt, A., Thomae, E., Bulygin, S., Wolf, C.: Small Public Keys and Fast Verification for \(\mathcal{M}\)ultivariate \(\mathcal{Q}\)uadratic Public Key Systems. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 475–490. Springer, Heidelberg (2011)
Pierce, R.S.: Associative Algebras, p. 16. Springer (1982)
Thomae, E.: A Generalization of the Rainbow Band Separation Attack and its Applications to Multivariate Schemes (2012), http://eprint.iacr.org/2012/223
Thomae, E., Wolf, C.: Solving Underdetermined Systems of Multivariate Quadratic Equations Revisited. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 156–171. Springer, Heidelberg (2012)
Yasuda, T., Sakurai, K., Takagi, T.: Reducing the Key Size of Rainbow Using Non-commutative Rings. In: Dunkelman, O. (ed.) CT-RSA 2012. LNCS, vol. 7178, pp. 68–83. Springer, Heidelberg (2012)
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Thomae, E. (2012). Quo Vadis Quaternion? Cryptanalysis of Rainbow over Non-commutative Rings. In: Visconti, I., De Prisco, R. (eds) Security and Cryptography for Networks. SCN 2012. Lecture Notes in Computer Science, vol 7485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32928-9_20
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DOI: https://doi.org/10.1007/978-3-642-32928-9_20
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