Skip to main content
SpringerLink
Log in

Extension Field Cancellation: A New Central Trapdoor for Multivariate Quadratic Systems

  • Conference paper
  • First Online:
Post-Quantum Cryptography (PQCrypto 2016)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9606))

Included in the following conference series:

Abstract

This paper introduces a new central trapdoor for multivariate quadratic (MQ) public-key cryptosystems that allows for encryption, in contrast to time-tested MQ primitives such as Unbalanced Oil and Vinegar or Hidden Field Equations which only allow for signatures. Our construction is a mixed-field scheme that exploits the commutativity of the extension field to dramatically reduce the complexity of the extension field polynomial implicitly present in the public key. However, this reduction can only be performed by the user who knows concise descriptions of two simple polynomials, which constitute the private key. After applying this transformation, the plaintext can be recovered by solving a linear system. We use the minus and projection modifiers to inoculate our scheme against known attacks. A straightforward C++ implementation confirms the efficient operation of the public key algorithms.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. NESSIE: New European Schemes for Signatures, Integrity, Encryption 05 November 2014 (2003). https://www.cosic.esat.kuleuven.be/nessie/

  2. Bettale, L., Faugère, J., Perret, L.: Cryptanalysis of HFE, multi-HFE and variants for odd and even characteristic. Des. Codes Crypt. 69(1), 1–52 (2013). http://dx.doi.org/10.1007/s10623-012-9617-2

    Article  MATH  Google Scholar 

  3. Billet, O., Patarin, J., Seurin, Y.: Analysis of intermediate field systems. IACR Cryptology ePrint Archive 2009, p. 542 (2009). http://eprint.iacr.org/2009/542

  4. Courtois, N.T.: The security of Hidden Field Equations (HFE). In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 266–281. Springer, Heidelberg (2001). http://dx.doi.org/10.1007/3-540-45353-9_20

    Chapter  Google Scholar 

  5. 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). http://dx.doi.org/10.1007/978-3-642-22792-9_41

    Chapter  Google Scholar 

  6. Ding, J., Kleinjung, T.: Degree of regularity for HFE. In: IACR Cryptology ePrint Archive 2011, p. 570 (2011). http://eprint.iacr.org/2011/570

  7. 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). http://dx.doi.org/10.1007/11496137_12

    Chapter  Google Scholar 

  8. 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). http://dx.doi.org/10.1007/978-3-642-38616-9_4

    Chapter  Google Scholar 

  9. Ding, J., Yang, B., Cheng, C., Chen, C.O., Dubois, V.: Breaking the symmetry: a way to resist the new differential attack. IACR Cryptology ePrint Archive 2007, p. 366 (2007). http://eprint.iacr.org/2007/366

  10. 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). http://dx.doi.org/10.1007/978-3-540-74143-5_1

    Chapter  Google Scholar 

  11. Dubois, V., Fouque, P.-A., Stern, J.: Cryptanalysis of SFLASH with slightly modified parameters. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 264–275. Springer, Heidelberg (2007). http://dx.doi.org/10.1007/978-3-540-72540-4_15

    Chapter  Google Scholar 

  12. Faugere, J.C.: A new efficient algorithm for computing gröbner bases (f 4). J. Pure Appl. Algebra 139(1), 61–88 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Faugère, J.C.: A new efficient algorithm for computing gröbner bases without reduction to zero (f5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ISSAC 2002, pp. 75–83. ACM, New York (2002). http://doi.acm.org/10.1145/780506.780516

  14. 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). http://dx.doi.org/10.1007/978-3-540-45146-4_3

    Chapter  Google Scholar 

  15. Gaborit, P. (ed.): PQCrypto 2013. LNCS, vol. 7932. Springer, Heidelberg (2013). http://dx.doi.org/10.1007/978-3-642-38616-9

    MATH  Google Scholar 

  16. Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998). http://dx.doi.org/10.1007/BFb0054868

    Chapter  Google Scholar 

  17. Kipnis, A., Patarin, J., Goubin, L.: Unbalanced oil and vinegar signature schemes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592. Springer, Heidelberg (1999). http://dx.doi.org/10.1007/3-540-48910-X_15

    Google Scholar 

  18. 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). http://dx.doi.org/10.1007/3-540-48405-1_2

    Google Scholar 

  19. 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). http://dx.doi.org/10.1007/3-540-45961-8_39

    Google Scholar 

  20. McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. DSN Prog. Rep. 42(44), 114–116 (1978)

    Google Scholar 

  21. 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). http://dx.doi.org/10.1007/3-540-44750-4_20

    Google Scholar 

  22. 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). http://dx.doi.org/10.1007/3-540-68339-9_4

    Google Scholar 

  23. 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). http://dx.doi.org/10.1007/3-540-49649-1_4

    Google Scholar 

  24. Porras, J., Baena, J., Ding, J.: ZHFE, a new multivariate public key encryption scheme. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 229–245. Springer, Heidelberg (2014). http://dx.doi.org/10.1007/978-3-319-11659-4_14

    Google Scholar 

  25. Shamir, A.: Efficient signature schemes based on birational permutations. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 1–12. Springer, Heidelberg (1994). http://dx.doi.org/10.1007/3-540-48329-2_1

    Google Scholar 

  26. 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). http://dx.doi.org/10.1007/978-3-642-12929-2_1

    Chapter  Google Scholar 

  27. Tao, C., Diene, A., Tang, S., Ding, J.: Simple matrix scheme for encryption. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 231–242. Springer, Heidelberg (2013). http://dx.doi.org/10.1007/978-3-642-38616-9_16

    Chapter  Google Scholar 

  28. Thomae, E.: About the security of multivariate quadratic public key schemes. Ph.D. thesis, Ruhr-Universität Bochum (2013)

    Google Scholar 

  29. Thomae, E., Wolf, C.: Cryptanalysis of enhanced TTS, STS and all its variants, or: why cross-terms are important. In: Mitrokotsa, A., Vaudenay, S. (eds.) AFRICACRYPT 2012. LNCS, vol. 7374, pp. 188–202. Springer, Heidelberg (2012). http://dx.doi.org/10.1007/978-3-642-31410-0_12

    Chapter  Google Scholar 

  30. Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theor. 32(1), 54–62 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wolf, C., Braeken, A., Preneel, B.: On the security of stepwise triangular systems. Des. Codes Crypt. 40(3), 285–302 (2006). http://dx.doi.org/10.1007/s10623-006-0015-5

    Article  MathSciNet  MATH  Google Scholar 

  32. Wolf, C., Preneel, B.: Taxonomy of public key schemes based on the problem of multivariate quadratic equations. In: IACR Cryptology ePrint Archive 2005, p. 77 (2005). http://eprint.iacr.org/2005/077

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful comments. This work was supported in part by the Research Council KU Leuven: C16/15/058. In addition, this work was supported by the Flemish Government, FWO WET G.0213.11N and by the European Commission through the ICT programme under contract FP7-ICT-2011-284833 PUFFIN, FP7-ICT-2013-10-SEP-210076296 PRACTICE, through the Horizon 2020research and innovation programme under grant agreement No H2020-ICT-2014-644371 WITDOM and H2020-ICT-2014-645622 PQCRYPTO; as well as by grant USDC (NIST) 60NAN15D059 from the Nation Institute of Standards of Technology. Alan Szepieniec is funded by a research grant of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, ESAT/COSIC, KU Leuven, Leuven, Belgium

    Alan Szepieniec & Bart Preneel

  2. iMinds, Ghent, Belgium

    Alan Szepieniec & Bart Preneel

  3. University of Cincinnati, Cincinnati, OH, USA

    Jintai Ding

Authors
  1. Alan Szepieniec
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jintai Ding
    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

Corresponding author

Correspondence to Alan Szepieniec .

Editor information

Editors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Tsuyoshi Takagi

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Szepieniec, A., Ding, J., Preneel, B. (2016). Extension Field Cancellation: A New Central Trapdoor for Multivariate Quadratic Systems. 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_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-29360-8_12

  • 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)

Publish with us

Policies and ethics

Search

Navigation