Skip to main content
Springer Nature Link
Log in

A Study of the Security of Unbalanced Oil and Vinegar Signature Schemes

  • Conference paper
Topics in Cryptology – CT-RSA 2005 (CT-RSA 2005)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 3376))

Included in the following conference series:

Abstract

The Unbalanced Oil and Vinegar scheme (UOV) is a signature scheme based on multivariate quadratic equations. It uses m equations and n variables. A total of v of these are called “vinegar variables”. In this paper, we study its security from several points of view. First, we are able to demonstrate that the constant part of the affine transformation does not contribute to the security of UOV and should therefore be omitted. Second, we show that the case n ≥ 2m is particularly vulnerable to Gröbner basis attacks. This is a new result for UOV over fields of odd characteristic. In addition, we investigate a modification proposed by the authors of UOV, namely to chose coefficients from a small subfield. This leads to a smaller public key. But due to the smaller key-space, this modification is insecure and should therefore be avoided. Finally, we demonstrate a new attack which works well for the case of small v. It extends the affine approximation attack from Youssef and Gong against the Imai-Matsumoto Scheme B for odd characteristic and applies it against UOV. This way, we point out serious vulnerabilities in UOV which have to be taken into account when constructing signature schemes based on UOV.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Computational Algebra Group, University of Sydney. The MAGMA Computational Algebra System for Algebra, Number Theory and Geometry, http://magma.maths.usyd.edu.au/magma/

  2. Courtois, N., Goubin, L., Meier, W., Tacier, J.-D.: Solving underdefined systems of multivariate quadratic equations. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 211–227. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  3. Courtois, N., Goubin, L., Patarin, J.: Quartz: Primitive specification (second revised version) v21- b. zip, 18 pages (October 2001), https://www.cosic.esat.kuleuven.ac.be/nessie/workshop/submissions/quartz

  4. Courtois, N., Goubin, L., Patarin, J.: SFlashv3, a fast asymmetric signature scheme – Revised Specificatoin of SFlash, version 3.0, 14 pages (October 17, 2003), ePrint Report 2003/211, http://eprint.iacr.org/

  5. Courtois, N.T., Daum, M., Felke, P.: On the security of HFE, HFEv- and Quartz. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 337–350. Springer, Heidelberg (2002), http://eprint.iacr.org/2002/138

    Chapter  Google Scholar 

  6. Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases (F 4). Journal of Pure and Applied Algebra 139, 61–88 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5). In: International Symposium on Symbolic and Algebraic Computation – ISSAC 2002, July 2002, pp. 75–83. ACM Press, New York (2002)

    Chapter  Google Scholar 

  8. Faugère, J.-C., Joux, A.: Algebraic cryptanalysis of Hidden Field Equations (HFE) using gröbner bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  9. Garay, M.R., Johnson, D.S.: Computers and Intractability – A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979) ISBN 0- 7167-1044-7 or 0-7167-1045-5

    Google Scholar 

  10. 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)

    Chapter  Google Scholar 

  11. Kasahara, M., Sakai, R.: A construction of public key cryptosystem for realizing ciphtertext of size 100 bit and digital signature scheme. IEICE Trans. Fundamentals E87-A(1), 102–109 (2004) Electronic version, http://search.ieice.org/2004/files/e000a01.htm#e87-a,1,102

  12. 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)

    Chapter  Google Scholar 

  13. Kipnis, A., Patarin, J., Goubin, L.: Unbalanced oil and vinegar signature schemes – extended version, 17 pages (2003) based on [12], citeseer/231623.html (2003- 06-11)

    Google Scholar 

  14. 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)

    Google Scholar 

  15. Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999); http://www.minrank.org/hfesubreg.ps or http://citeseer.nj.nec.com/kipnis99cryptanalysis.html

    Google Scholar 

  16. Matsumoto, T., Imai, H.: Algebraic methods for constructing asymmetric cryptosystems. In: Calmet, J. (ed.) AAECC 1985. LNCS, vol. 229, pp. 108–119. Springer, Heidelberg (1986)

    Google Scholar 

  17. 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 

  18. Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996), ISBN 0-8493-8523-7, online-version, http://www.cacr.math.uwaterloo.ca/hac/

  19. Moh, T.: A public key system with signature and master key function. Communications in Algebra 27(5), 2207–2222 (1999), electronic version at http://citeseer/moh99public.html

    Google Scholar 

  20. Patarin, J.: Cryptanalysis of the matsumoto and imai public key scheme of eurocrypt 1988. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)

    Google Scholar 

  21. Patarin, J.: Asymmetric cryptography with a hidden monomial. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 45–60. Springer, Heidelberg (1996)

    Google Scholar 

  22. Patarin, J.: Hidden Field 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), http://www.minrank.org/hfe.pdf

    Chapter  Google Scholar 

  23. Patarin, J.: The oil and vinegar signature scheme. Presented at the Dagstuhl Workshop on Cryptography (September 1997) (transparencies)

    Google Scholar 

  24. Patarin, J., Goubin, L.: Trapdoor one-way permutations and multivariate polynomials. In: Han, Y., Quing, S. (eds.) ICICS 1997. LNCS, vol. 1334, pp. 356–368. Springer, Heidelberg (1997), http://citeseer.nj.nec.com/patarin97trapdoor.html

    Chapter  Google Scholar 

  25. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26(5), 1484–1509 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Toli, I.: Cryptanalysis of HFE, arXiv preprint server, 7 pages (June 2003), http://arxiv.org/abs/cs.CR/0305034

  27. Wolf, C., Braeken, A., Preneel, B.: Efficient cryptanalysis of RSE(2)PKC and RSSE(2)PKC. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, p. 14. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  28. Yang, B.-Y., Chen, J.-M.: Rank attacks and defence in Tame-like multivariate PKC’s. Cryptology ePrint Archive, Report 2004/061, 21 pages (March 23, 2004). http://eprint.iacr.org/

    Google Scholar 

  29. Youssef, A.M., Gong, G.: Cryptanalysis of Imai and Matsumoto scheme B asymmetric cryptosystem. In: Pandu Rangan, C., Ding, C. (eds.) INDOCRYPT 2001. LNCS, vol. 2247, pp. 214–222. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. K.U.Leuven, ESAT-COSIC, Kasteelpark Arenberg 10, B-3001, Leuven-Heverlee, Belgium

    An Braeken, Christopher Wolf & Bart Preneel

Authors
  1. An Braeken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christopher Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bart Preneel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Combinatorics & Optimization, University of Waterloo,  

    Alfred Menezes

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Braeken, A., Wolf, C., Preneel, B. (2005). A Study of the Security of Unbalanced Oil and Vinegar Signature Schemes. In: Menezes, A. (eds) Topics in Cryptology – CT-RSA 2005. CT-RSA 2005. Lecture Notes in Computer Science, vol 3376. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30574-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-30574-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24399-1

  • Online ISBN: 978-3-540-30574-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics