Advertisement

Linear Recurring Sequences for the UOV Key Generation Revisited

  • Albrecht Petzoldt
  • Stanislav Bulygin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7839)

Abstract

Multivariate cryptography is one of the main candidates to guarantee the security of communication in a post quantum era. While multivariate signature schemes are very fast and require only modest computational resources, the key sizes of such schemes are quite large. In [17] Petzoldt et al. proposed a way to use Linear Recurring Sequences (LRS’s) for the key generation of the Unbalanced Oil and Vinegar (UOV) signature scheme by which they were able to reduce the public key size of this scheme by a factor of 7. In this paper we describe a modification of their scheme, which enables us not only to reduce the public key size, but also to speed up the verification process of the UOV scheme by a factor of 5.

Keywords

Multivariate Cryptography UOV Signature Scheme Key Size Reduction Fast Verification 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.): Post-Quantum Cryptography. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  2. [2]
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    Bogdanov, A., Eisenbarth, T., Rupp, A., Wolf, C.: Time-area optimized public-key engines: -cryptosystems as replacement for elliptic curves? In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 45–61. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. [4]
    Berbain, C., Gilbert, H., Patarin, J.: QUAD: A Practical Stream Cipher with Provable Security. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 109–128. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. [5]
    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)CrossRefGoogle Scholar
  6. [6]
    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)CrossRefGoogle Scholar
  7. [7]
    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)CrossRefGoogle Scholar
  8. [8]
    Ding, J., Wolf, C., Yang, B.-Y.: ℓ-invertible Cycles for Multivariate Quadratic Public Key Cryptography. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 266–281. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. [9]
    Faugère, J.C.: A new efficient algorithm for computing Groebner bases (F4). Journal of Pure and Applied Algebra 139, 61–88 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company (1979)Google Scholar
  11. [11]
    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)CrossRefGoogle Scholar
  12. [12]
    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)CrossRefGoogle Scholar
  13. [13]
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and their Applications. Cambridge University Press (1986)Google Scholar
  14. [14]
    Matsumoto, T., Imai, H.: Public Quadratic Polynomial-Tuples for efficient Signature-Verification and Message-Encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  15. [15]
    Patarin, J.: The oil and vinegar signature scheme. Presented at the Dagstuhl Workshop on Cryptography (September 1997)Google Scholar
  16. [16]
    Petzoldt, A., Bulygin, S., Buchmann, J.: A Multivariate Signature Scheme with a partially cyclic public key. In: Proceedings of SCC 2010, pp. 229–235 (2010)Google Scholar
  17. [17]
    Petzoldt, A., Bulygin, S., Buchmann, J.: Linear Recurring Sequences for the UOV Key Generation. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 335–350. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. [18]
    Shor, P.: Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM J. Comput. 26(5), 1484–1509Google Scholar
  19. [19]
    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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Albrecht Petzoldt
    • 1
  • Stanislav Bulygin
    • 2
  1. 1.Department of Computer ScienceTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Center for Advanced Security Research Darmstadt - CASEDDarmstadtGermany

Personalised recommendations