CyclicRainbow – A Multivariate Signature Scheme with a Partially Cyclic Public Key

  • Albrecht Petzoldt
  • Stanislav Bulygin
  • Johannes Buchmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6498)


Multivariate Cryptography is one of the alternatives to guarantee the security of communication in the post-quantum world. One major drawback of such schemes is the huge size of their keys. In [PB10] Petzoldt et al. proposed a way how to reduce the public key size of the UOV scheme by a large factor. In this paper we extend this idea to the Rainbow signature scheme of Ding and Schmidt [DS05]. By our construction it is possible to reduce the size of the public key by up to 62%.


Multivariate Cryptography Rainbow Signature Scheme Key Size Reduction 


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  1. [BB08]
    Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.): Post Quantum Cryptography. Springer, Heidelberg (2009)MATHGoogle Scholar
  2. [BC97]
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. [BF09]
    Bettale, L., Faugere, J.-C., Perret, L.: Hybrid approach for solving multivariate systems over finite fields. Journal of Math. Cryptology, 177–197 (2009)Google Scholar
  4. [BG06]
    Billet, O., Gilbert, H.: Cryptanalysis of Rainbow. In: De Prisco, R., Yung, M. (eds.) SCN 2006. LNCS, vol. 4116, pp. 336–347. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. [DS05]
    Ding, J., Schmidt, D.: Rainbow, a new multivariate 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
  6. [Di04]
    Ding, J.: A new variant of the Matsumoto-Imai cryptosystem through perturbation. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 266–281. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. [DY08]
    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. [DW07]
    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. [DY07]
    Ding, J., Yang, B.-Y., Cheng, C.-M., Chen, O., Dubois, V.: Breaking the symmetry: a Way to Resist the new Differential Attack, eprint 366/2007Google Scholar
  10. [Fa99]
    Faugére, J.C.: A new efficient algorithm for computing Groebner bases (F4). Journal of Pure and Applied Algebra 139, 61–88 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. [FP09]
    Faugére, J.C., Perret, L.: An efficient algorithm for decomposing multivariate polynomials and its applications to cryptography. Journal of Symbolic Computation 44(12), 1676–1689 (2009)MATHCrossRefMathSciNetGoogle Scholar
  12. [GC00]
    Goubin, L., Courtois, N.T.: Cryptanalysis of the TTM cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. [HW05]
    Hu, Y.-H., Wang, L.-C., Chou, C.-P., Lai, F.: Similar Keys of Multivariate Public Key Cryptosystems. In: Desmedt, Y.G., Wang, H., Mu, Y., Li, Y. (eds.) CANS 2005. LNCS, vol. 3810, pp. 211–222. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. [KP99]
    Kipnis, A., Patarin, L., Goubin, L.: Unbalanced Oil and Vinegar Schemes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 206–222. Springer, Heidelberg (1999)Google Scholar
  15. [KS98]
    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)Google Scholar
  16. [MI88]
    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)Google Scholar
  17. [Pa96]
    Patarin, J.: Hidden Field equations (HFE) and Isomorphisms of Polynomials (IP). In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 38–48. Springer, Heidelberg (1996)Google Scholar
  18. [Pa97]
    Patarin, J.: The oil and vinegar signature scheme, presented at the Dagstuhl Workshop on Cryptography (September 1997)Google Scholar
  19. [PB10]
    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
  20. [PB1a]
    Petzoldt, A., Bulygin, S., Buchmann, J.: Selecting Parameters for the Rainbow Signature Scheme. In: Sendrier, N. (ed.) Post-Quantum Cryptography. LNCS, vol. 6061, pp. 218–240. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  21. [PC01]
    Patarin, J., Courtois, N., Goubin, L.: Flash, a fast multivariate signature algorithm. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 298–307. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. [PG98]
    Patarin, J., Goubin, L., Courtois, N.: 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)CrossRefGoogle Scholar
  23. [TG10]
    Tsuji, S., Gotaishi, M., Tadaki, K., Fujita, R.: Proposal of a Signature Scheme based on STS Trapdoor. In: Sendrier, N. (ed.) Post-Quantum Cryptography. LNCS, vol. 6061, pp. 201–217. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  24. [WY06]
    Wang, L.C., Yang, B.Y., Hu, Y.H., Lai, F.: A medium field multivariate public-key encryption scheme. In: Pointcheval, D. (ed.) CT-RSA 2006. LNCS, vol. 3860, pp. 132–149. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  25. [YC05]
    Yang, B.-Y., Chen, J.-M.: Building secure tame like multivariate public-key cryptosystems: The new TTS. In: Boyd, C., González Nieto, J.M. (eds.) ACISP 2005. LNCS, vol. 3574, pp. 518–531. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  26. [YC07]
    Yang, B.-Y., Chen, J.-M.: All in the XL family: Theory and practice. In: Park, C.-s., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 67–86. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

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

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