Abstract
ZHFE, proposed by Porras et al. at PQCrypto’14, is one of the very few existing multivariate encryption schemes and a very promising candidate for post-quantum cryptosystems. The only one drawback is its slow key generation. At PQCrypto’16, Baena et al. proposed an algorithm to construct the private ZHFE keys, which is much faster than the original algorithm, but still inefficient for practical parameters. Recently, Zhang and Tan proposed another private key generation algorithm, which is very fast but not necessarily able to generate all the private ZHFE keys. In this paper we propose a new efficient algorithm for the private key generation of the ZHFE scheme. Our algorithm reduces the complexity from \(O(n^{2\omega +1})\) by Baena et al. to \(O(n^{\omega +3})\), where n is the number of variables and \(2<\omega <3\) is a linear algebra constant. We also estimate the number of possible keys generated by all existing private key generation algorithms for ZHFE. Our algorithm generates as many private ZHFE keys as the original and Baena et al.’s ones. This makes our algorithm be the best appropriate for the ZHFE scheme.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Here we used the Magma’s command \(\texttt {GetMaximumMemoryUsage}\) to measure max memory. Note also that we used the Magma’s command \(\texttt {Solution}\) to solve linear systems in the algorithm.
References
Bernstein, D.J., Buchmann, J., Dahmen, E.: Post-Quantum Cryptography. Springer, Heidelberg (2009)
Baena, J.B., Cabarcas, D., Escudero, D.E., Porras-Barrera, J., Verbel, J.A.: Efficient ZHFE key generation. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 213–232. Springer, Cham (2016). doi:10.1007/978-3-319-29360-8_14
Bogdanov, A., Eisenbarth, T., Rupp, A., Wolf, C.: Time-area optimized public-key engines: \(\cal{MQ}\)-cryptosystems as replacement for elliptic curves? In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 45–61. Springer, Heidelberg (2008). doi:10.1007/978-3-540-85053-3_4
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symbolic Comput. 24(3–4), 235–265 (1997)
Chen, A.I.-T., Chen, M.-S., Chen, T.-R., Cheng, C.-M., Ding, J., Kuo, E.L.-H., Lee, F.Y.-S., Yang, B.-Y.: SSE implementation of multivariate PKCs on modern x86 CPUs. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 33–48. Springer, Heidelberg (2009). doi:10.1007/978-3-642-04138-9_3
Courtois, N.T.: The security of hidden field equations (HFE). In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 266–281. Springer, Heidelberg (2001). doi:10.1007/3-540-45353-9_20
Ding, J., Gower, J.E., Schmidt, D.S.: Multivariate Public Key Cryptosystems. Springer, Heidelberg (2006)
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). doi:10.1007/11496137_12
Goodin, D.: NSA preps quantum-resistant algorithms to head off crypto-apocalypse. http://arstechnica.com/security/2015/08/nsa-preps-quantum-resistant-al-gorithms-to-head-off-crypto-apocolypse/
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H Freeman and Company, New York (1979)
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). doi:10.1007/3-540-48910-X_15
Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem by relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999). doi:10.1007/3-540-48405-1_2
Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In: Barstow, D., Brauer, W., Brinch Hansen, P., Gries, D., Luckham, D., Moler, C., Pnueli, A., Seegmüller, G., Stoer, J., Wirth, N., Günther, C.G. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988). doi:10.1007/3-540-45961-8_39
National Institute of Standards and Technology: Report on Post Quantum Cryptography, NISTIR draft 8105. http://csrc.nist.gov/publications/drafts/nistir-8105/nistir_8105_draft.pdf
Patarin, J.: Cryptanalysis of the Matsumoto and Imai public key scheme of eurocrypt 88. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995). doi:10.1007/3-540-44750-4_20
Patarin, J.: Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996). doi:10.1007/3-540-68339-9_4
Perlner, R., Smith-Tone, D.: Security analysis and key modification for ZHFE. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 197–212. Springer, Heidelberg (2016). doi:10.1007/978-3-319-29360-8_13
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). doi:10.1007/978-3-662-48797-6_14
Porras, J., Baena, J., Ding, J.: New candidates for multivariate trapdoor functions. Cryptology ePrint Archive, Report 2014/387 (2014)
Porras, J., Baena, J., Ding, J.: ZHFE, a new multivariate public key encryption scheme. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 229–245. Springer, Heidelberg (2014). doi:10.1007/978-3-319-11659-4_14
Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)
Tao, C., Diene, A., Tang, S., Ding, J.: Simple matrix scheme for encryption. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 231–242. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38616-9_16
Szepieniec, A., Ding, J., Preneel, B.: Extension field cancellation: a new central trapdoor for multivariate quadratic systems. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 182–196. Springer, Cham (2016). doi:10.1007/978-3-319-29360-8_12
Zhang, W., Tan, C.H.: On the Security and key generation of the ZHFE encryption scheme. In: Ogawa, K., Yoshioka, K. (eds.) IWSEC 2016. LNCS, vol. 9836, pp. 289–304. Springer, Heidelberg (2016). doi:10.1007/978-3-319-44524-3_17
Yasuda, T., Sakurai, K.: A multivariate encryption scheme with rainbow. In: Qing, S., Okamoto, E., Kim, K., Liu, D. (eds.) ICICS 2015. LNCS, vol. 9543, pp. 236–251. Springer, Heidelberg (2016). doi:10.1007/978-3-319-29814-6_19
Acknowledgments
This work was supported by CREST, JST. The second author also acknowledges the Japanese Society for the Promotion of Science (JSPS) for financial support under grant KAKENHI 16K17644.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
A Our Algorithm in Sect. 3.3
A Our Algorithm in Sect. 3.3
The expression \(f\xleftarrow {R} W\) denotes that f is an element chosen uniformly at random from the set W.
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Ikematsu, Y., Duong, D.H., Petzoldt, A., Takagi, T. (2017). Revisiting the Efficient Key Generation of ZHFE. In: El Hajji, S., Nitaj, A., Souidi, E. (eds) Codes, Cryptology and Information Security. C2SI 2017. Lecture Notes in Computer Science(), vol 10194. Springer, Cham. https://doi.org/10.1007/978-3-319-55589-8_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-55589-8_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-55588-1
Online ISBN: 978-3-319-55589-8
eBook Packages: Computer ScienceComputer Science (R0)