Abstract
Multivariate Public Key Cryptography (MPKC) has been put forth as a possible post-quantum family of cryptographic schemes. These schemes lack provable security in the reduction theoretic sense, and so their security against yet undiscovered attacks remains uncertain. The effectiveness of differential attacks on various field-based systems has prompted the investigation of differential properties of multivariate schemes to determine the extent to which they are secure from differential adversaries. Due to its role as a basis for both encryption and signature schemes we contribute to this investigation focusing on the HFE cryptosystem. We derive the differential symmetric and invariant structure of the HFE central map and that of HFE − and provide a collection of parameter sets which make these HFE systems provably secure against a differential symmetric or differential invariant attack.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Sci. Stat. Comp. 26, 1484 (1997)
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.A., Smith-Tone, D.: A classification of differential invariants for multivariate post-quantum cryptosystems. In: [24], pp. 165–173
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)
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)
Patarin, J.: Cryptoanalysis 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)
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)
Patarin, J., Goubin, L., Courtois, N.T.: C \(^*_{-+}\) and HM: Variations around two schemes of T.Matsumoto and H.Imai. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 35–50. Springer, Heidelberg (1998)
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)
Ding, J., Kleinjung, T.: Degree of regularity for hfe-. IACR Cryptology ePrint Archive 2011, 570 (2011)
Ding, J., Yang, B.Y.: Degree of regularity for hfev and hfev-. In: [24] pp. 52–66
Bettale, L., Faugère, J.C., Perret, L.: Cryptanalysis of hfe, multi-hfe and variants for odd and even characteristic. Des. Codes Cryptography 69(1), 1–52 (2013)
Granboulan, L., Joux, A., Stern, J.: Inverting hfe is quasipolynomial. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 345–356. Springer, Heidelberg (2006)
Patarin, J.: The oil and vinegar algorithm for signatures. Presented at the Dagsthul Workshop on Cryptography (1997)
Moody, D., Perlner, R.A., 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)
Matsumoto, T., Imai, H.: Public Quadratic Polynominal-Tuples for Efficient Signature-Verification and Message-Encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988)
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)
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)
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)
Bouillaguet, C., Fouque, P.A., Joux, A., Treger, J.: A family of weak keys in hfe and the corresponding practical key-recovery. J. Mathematical Cryptology 5, 247–275 (2012)
Wolf, C., Preneel, B.: Equivalent keys in multivariate quadratic public key systems. J. Mathematical Cryptology 4, 375–415 (2011)
Ding, J., Hodges, T.J.: Inverting hfe systems is quasi-polynomial for all fields. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 724–742. Springer, Heidelberg (2011)
Bardet, M., Faugere, J.C., Salvy, B.: On the complexity of gröbner basis computation of semi-regular overdetermined algebraic equations. In: Proceedings of the International Conference on Polynomial System Solving (2004)
Gaborit, P. (ed.): PQCrypto 2013. LNCS, vol. 7932. Springer, Heidelberg (2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Daniels, T., Smith-Tone, D. (2014). Differential Properties of the HFE Cryptosystem. In: Mosca, M. (eds) Post-Quantum Cryptography. PQCrypto 2014. Lecture Notes in Computer Science, vol 8772. Springer, Cham. https://doi.org/10.1007/978-3-319-11659-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-11659-4_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11658-7
Online ISBN: 978-3-319-11659-4
eBook Packages: Computer ScienceComputer Science (R0)