Abstract
Multivariate Public Key Cryptography (MPKC) is one of the most attractive post-quantum options for digital signatures in a wide array of applications. The history of multivariate signature schemes is tumultuous, however, and solid security arguments are required to inspire faith in the schemes and to verify their security against yet undiscovered attacks. The effectiveness of “differential attacks” on various field-based systems has prompted the investigation of the resistance of schemes against differential adversaries. Due to its prominence in the area and the recent optimization of its parameters, we prove the security of \(HFEv^-\) against differential adversaries. We investigate the newly suggested parameters and conclude that the proposed scheme is secure against all known attacks and against any differential adversary.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Lange, T., et al.: Post-quantum cryptography for long term security. Horizon 2020, ICT-645622 (2015) http://cordis.europa.eu/project/rcn/194347_en.html
Campagna, M., Chen, L., et al.: Quantum safe cryptography and security. ETSI White Paper No. 8 (2015). http://www.etsi.org/images/files/ETSIWhitePapers/QuantumSafeWhitepaper.pdf
Moody, D., Chen, L., Liu, Y.K.: Nist pqc workgroup. Computer Security Resource Center (2015). http://csrc.nist.gov/groups/ST/crypto-research-projects/#PQC
Smith-Tone, D.: On the differential security of multivariate public key cryptosystems. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 130–142. Springer, Heidelberg (2011)
Perlner, R., Smith-Tone, D.: A classification of differential invariants for multivariate post-quantum cryptosystems. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 165–173. Springer, Heidelberg (2013)
Daniels, T., Smith-Tone, D.: Differential properties of the HFE cryptosystem. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 59–75. Springer, Heidelberg (2014)
Dubois, V., Fouque, P.-A., Shamir, A., Stern, J.: Practical cryptanalysis of SFLASH. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 1–12. Springer, Heidelberg (2007)
Shamir, A., Kipnis, A.: Cryptanalysis of the oil & vinegar signature scheme. In: Krawczyk, H. (ed.) CRYPTO ’98. LNCS, pp. 257–266. Springer, Heidelberg (1998)
Moody, D., Perlner, R., Smith-Tone, D.: An asymptotically optimal structural attack on the ABC multivariate encryption scheme. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 180–196. Springer, Heidelberg (2014)
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)
Perlner, R., Smith-Tone, D.: Security analysis and key modification for ZHFE. In: Post-Quantum Cryptography - 7th International Conference, PQCrypto 2016, 24–26 February 2016, Fukuoka, Japan (2016)
Patarin, J., Courtois, N.T., Goubin, L.: QUARTZ, 128-bit long digital signatures. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 282–297. Springer, Heidelberg (2001)
Petzoldt, A., Chen, M., Yang, B., 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)
Patarin, J.: Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996)
Matsumoto, T., Imai, H.: Public quadratic polynominal-tuples for efficient signature-verification and message-encryption. In: EUROCRYPT, pp. 419–453 (1988)
Berlekamp, E.R.: Factoring polynomials over large finite fields. Math. Comput. 24, 713–735 (1970)
Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem by relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, p. 19. Springer, Heidelberg (1999)
Bettale, L., Faugère, J., Perret, L.: Cryptanalysis of HFE, multi-HFE and variants for odd and even characteristic. Des. Codes Crypt. 69, 1–52 (2013)
Fouque, P.-A., Macario-Rat, G., Perret, L., Stern, J.: Total break of the \(\ell \)-IC signature scheme. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 1–17. Springer, Heidelberg (2008)
Smith-Tone, D.: Properties of the discrete differential with cryptographic applications. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 1–12. Springer, Heidelberg (2010)
Faugère, J.-C., Joux, A.: Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using gröbner bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003)
Ding, J., Yang, B.-Y.: Degree of regularity for HFEv and HFEv-. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 52–66. Springer, Heidelberg (2013)
Wolf, C., Preneel, B.: Equivalent keys in multivariate quadratic public key systems. J. Math. Crypt. 4, 375–415 (2011)
Gaborit, P. (ed.): PQCrypto 2013. Security and Cryptology, vol. 7932. Springer, Heidelberg (2013)
Mosca, M. (ed.): PQCrypto 2014. LNCS, vol. 8772. Springer, Heidelberg (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Cartor, R., Gipson, R., Smith-Tone, D., Vates, J. (2016). On the Differential Security of the HFEv- Signature Primitive. In: Takagi, T. (eds) Post-Quantum Cryptography. PQCrypto 2016. Lecture Notes in Computer Science(), vol 9606. Springer, Cham. https://doi.org/10.1007/978-3-319-29360-8_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-29360-8_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29359-2
Online ISBN: 978-3-319-29360-8
eBook Packages: Computer ScienceComputer Science (R0)