Abstract
One of the most efficient post-quantum signature schemes is Rainbow whose hardness is based on the multivariate quadratic polynomial (MQ) problem. ELSA, a new multivariate signature scheme proposed at Asiacrypt 2017, has a similar construction to Rainbow. Its advantages, compared to Rainbow, are its smaller secret key and faster signature generation. In addition, its existential unforgeability against an adaptive chosen-message attack has been proven under the hardness of the MQ-problem induced by a public key of ELSA with a specific parameter set in the random oracle model. The high efficiency of ELSA is derived from a set of hidden quadratic equations used in the process of signature generation. However, the hidden quadratic equations yield a vulnerability. In fact, a piece of information of these equations can be recovered by using valid signatures and an equivalent secret key can be partially recovered from it. In this paper, we describe how to recover an equivalent secret key of ELSA by a chosen message attack. Our experiments show that we can recover an equivalent secret key for the claimed 128-bit security parameter of ELSA on a standard PC in 177 s with 1326 valid signatures.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.): Post-Quantum Cryptography. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-540-88702-7
Bettale, L., Faugère, J.C., Perret, L.: Solving polynomial systems over finite fields: improved analysis of the hybrid approach. In: ISSAC 2012, pp. 67–74 (2012)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)
Ding, J., Chen, M.C., Petzoldt, A., Schmidt, D., Yang, B.Y.: Rainbow, NIST, Post-Quantum Cryptography Standardization, Round 1 Submissions. https://csrc.nist.gov/Projects/Post-Quantum-Cryptography/Round-1-Submissions
Ding, J., Gower, J.E., Schmidt, D.S.: Multivariate Public Key Cryptosystems. Springer, Boston (2006). https://doi.org/10.1007/978-0-387-36946-4
Ding, J., Schmidt, D.: Rainbow, a new multivariable polynomial signature scheme. In: Ioannidis, J., Keromytis, A., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005). https://doi.org/10.1007/11496137_12
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). https://doi.org/10.1007/3-540-48910-X_15
Kipnis, A., Shamir, A.: Cryptanalysis of the oil and vinegar signature scheme. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 257–266. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0055733
Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In: Barstow, D., et al. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988). https://doi.org/10.1007/3-540-45961-8_39
NIST, Post-Quantum Cryptography Standardization. https://csrc.nist.gov/Projects/Post-Quantum-Cryptography/
Patarin, J., Courtois, N., Goubin, L.: QUARTZ, 128-bit long digital signatures. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 282–297. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45353-9_21
Petzoldt, A., Chen, M.-S., Yang, B.-Y., Tao, C., Ding, J.: Design principles for HFEv-based multivariate signature schemes. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 311–334. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48797-6_14
Shim, K.-A., Park, C.-M., Koo, N.: An existential unforgeable signature scheme based on multivariate quadratic equations. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 37–64. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_2
Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)
Tsujii, S., Itoh, T., Fujioka, A., Kurosawa, K., Matsumoto, T.: A public-key cryptosystem based on the difficulty of solving a system of non-linear equations. Syst. Comput. Jpn. 19(2), 10–18 (1988)
Acknowledgements
This work was supported by JST CREST (Grant Number JPMJCR14D6). The first author was also supported by JSPS Grant-in-Aid for Scientific Research (C) no. 17K05181.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Hashimoto, Y., Ikematsu, Y., Takagi, T. (2018). Chosen Message Attack on Multivariate Signature ELSA at Asiacrypt 2017. In: Inomata, A., Yasuda, K. (eds) Advances in Information and Computer Security. IWSEC 2018. Lecture Notes in Computer Science(), vol 11049. Springer, Cham. https://doi.org/10.1007/978-3-319-97916-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-97916-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-97915-1
Online ISBN: 978-3-319-97916-8
eBook Packages: Computer ScienceComputer Science (R0)