Skip to main content
SpringerLink
Log in

Two-Face: New Public Key Multivariate Schemes

  • Conference paper
  • First Online:
Progress in Cryptology – AFRICACRYPT 2018 (AFRICACRYPT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10831))

Included in the following conference series:

Abstract

We present here new multivariate schemes that can be seen as HFE generalization having a property called ‘Two-Face’. Particularly, we present five such families of algorithms named ‘Dob’, ‘Simple Pat’, ‘General Pat’, ‘Mac’, and ‘Super Two-Face’. These families have connections between them, some of them are refinements or generalizations of others. Notably, some of these schemes can be used for public key encryption, and some for public key signature. We introduce also new multivariate quadratic permutations that may have interest beyond cryptography.

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 64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.00
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. Gilbert, H., Minier, M.: Cryptanalysis of SFLASH. [36], pp. 288–298 (2002)

    Google Scholar 

  2. Fouque, P., Macario-Rat, G., Stern, J.: Key recovery on hidden monomial multivariate schemes. [37], pp. 19–30 (2008)

    Google Scholar 

  3. Ding, J., Dubois, V., Yang, B., Chen, C.O., Cheng, C.: Could SFLASH be repaired? IACR Cryptology ePrint Archive 2009, 596 (2009)

    Google Scholar 

  4. Faugère, J., Perret, L.: On the security of UOV. IACR Cryptology ePrint Archive 2009, 483 (2009)

    Google Scholar 

  5. Hamdi, O., Bouallegue, A., Harari, S.: Hidden field equations cryptosystem performances. In: AICCSA, pp. 308–311. IEEE Computer Society (2006)

    Google Scholar 

  6. Patarin, J.: Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. [34], pp. 33–48 (1996)

    Google Scholar 

  7. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  8. Dembowski, P., Ostrom, T.G.: Planes of order \(n\) with collineation groups of order \(n^2\). Math. Z. 103(3), 239–258 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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). https://doi.org/10.1007/978-3-642-38616-9_4

    Chapter  Google Scholar 

  10. Dobbertin, H.: Almost perfect nonlinear power functions on GF(2\({}^{n}\)): the Welch case. IEEE Trans. Inf. Theory 45(4), 1271–1275 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Patarin, J.: Cryptanalysis of the Matsumoto and Imai public key scheme of eurocrypt’98. Des. Codes Crypt. 20(2), 175–209 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fouque, P., Granboulan, L., Stern, J.: Differential cryptanalysis for multivariate schemes. [32], pp. 341–353 (2005)

    Google Scholar 

  13. Dubois, V., Granboulan, L., Stern, J.: Cryptanalysis of HFE with internal perturbation. [33]. pp. 249–265 (2007)

    Google Scholar 

  14. Bouillaguet, C., Fouque, P.-A., Macario-Rat, G.: Practical key-recovery for all possible parameters of SFLASH. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 667–685. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_36

    Chapter  Google Scholar 

  15. 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). https://doi.org/10.1007/978-3-540-74143-5_1

    Chapter  Google Scholar 

  16. Salmon, G.: Lessons Introductory to the Modern Higher Algebra. Elibron Classics Series. Adegi Graphics LLC, Rye Brook (1999)

    Google Scholar 

  17. Geddes, K.O., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer Academic Publishers, Norwell (1992)

    Book  MATH  Google Scholar 

  18. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997). Computational algebra and number theory (London, 1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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). https://doi.org/10.1007/978-3-540-45146-4_3

    Chapter  Google Scholar 

  20. Bettale, L., Faugère, J.-C., Perret, L.: Cryptanalysis of multivariate and odd-characteristic HFE variants. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 441–458. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19379-8_27

    Chapter  Google Scholar 

  21. Bettale, L., Faugère, J., Perret, L.: Cryptanalysis of HFE, multi-HFE and variants for odd and even characteristic. IACR Cryptology ePrint Archive 2011, 399 (2011)

    Google Scholar 

  22. Billet, O., Patarin, J., Seurin, Y.: Analysis of intermediate field systems. In: First Conference on Symbolic Computation and Cryptography, Beijing, China, 28–30 April 2008, pp. 110–117 (2008)

    Google Scholar 

  23. Goubin, L., Courtois, N.: Cryptanalysis of the TTM cryptosystem. [24], pp. 44–57 (2000)

    Google Scholar 

  24. Okamoto, T. (ed.) Advances in Cryptology - ASIACRYPT 2000. LNCS, vol. 1976. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44448-3

  25. Zhang, W., Tan, C.H.: A new perturbed Matsumoto-Imai signature scheme. [26], pp. 43–48 (2014)

    Google Scholar 

  26. Emura, K., Hanaoka, G., Zhao, Y. (eds.): Proceedings of the 2nd ACM Workshop on ASIA Public-Key Cryptography, ASIAPKC 2014, 3 June, 2014, Kyoto, Japan. ACM (2014)

    Google Scholar 

  27. Zhang, W., Tan, C.H.: MI-T-HFE, a new multivariate signature scheme. Cryptology ePrint Archive, Report 2015/890 (2015). http://eprint.iacr.org/2015/890

  28. Ding, J., Gower, J.E., Schmidt, D., Wolf, C., Yin, Z.: Complexity estimates for the F4 attack on the perturbed Matsumoto-Imai cryptosystem. [29], pp. 262–277 (2005)

    Google Scholar 

  29. Smart, N.P. (ed.): Cryptography and Coding 2005. LNCS, vol. 3796. Springer, Heidelberg (2005). https://doi.org/10.1007/11586821

  30. Ding, J.: A new variant of the Matsumoto-Imai cryptosystem through perturbation. [31], pp. 305–318 (2004)

    Google Scholar 

  31. Bao, F., Deng, R.H., Zhou, J. (eds.): Public Key Cryptography-PKC 2004. LNCS, vol. 2947. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24632-9_22

  32. Cramer, R. (ed.): Advances in Cryptology - EUROCRYPT 2005. vol.3494. LNCS, Springer, Heidelberg (2005). https://doi.org/10.1007/11426639_20

  33. Okamoto, T., Wang, X. (eds.): Public Key Cryptography - PKC 2007. LNCS, vol. 4450. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71677-8_17

  34. Maurer, U.M. (ed.): Advances in Cryptology - EUROCRYPT 1996. LNCS, vol. 1070. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68339-9_4

  35. MacAulay, F.S.: Some formulæ in elimination. Proc. Lond. Math. Soc. s1–35(1), 3–27 (1902)

    Article  MATH  Google Scholar 

  36. Knudsen, L.R. (ed.): Advances in Cryptology - EUROCRYPT 2002. LNCS, vol. 2332. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-46035-7

  37. Smart, N.P. (ed.): Advances in Cryptology - EUROCRYPT 2008. LNCS, vol. 4965. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78967-3_2

  38. Hou, X.d.: Permutation polynomials over finite fields - a survey of recent advances. Finite Fields Appl. 32(C), 82–119 (2015)

    Google Scholar 

  39. Blokhuis, A., Coulter, R.S., Henderson, M., O’Keefe, C.M.: Permutations amongst the Dembowski-Ostrom polynomials. In: Jungnickel, D., Niederreiter, H. (eds.) Finite Fields and Applications, pp. 37–42. Springer, Heidelberg (2001). https://doi.org/10.1007/978-3-642-56755-1_4

  40. Plût, J., Fouque, P., Macario-Rat, G.: Solving the “isomorphism of polynomials with two secrets” problem for all pairs of quadratic forms. CoRR abs/1406.3163 (2014)

    Google Scholar 

Download references

Acknowledgements

We thank Ludovic Perret and Jean Charles Faugère, INRIA, for fruitful discussions and help for the experimental computations.

Author information

Authors and Affiliations

  1. Orange, Châtillon, France

    Gilles Macario-Rat

  2. Université Versailles Saint-Quentin, Versailles, France

    Jacques Patarin

Authors
  1. Gilles Macario-Rat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacques Patarin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilles Macario-Rat .

Editor information

Editors and Affiliations

  1. Fondation Partenariale de Sorbonne Université, Paris, France

    Antoine Joux

  2. Université de Caen, Caen, France

    Abderrahmane Nitaj

  3. Al Akhawayn University, Ifrane, Morocco

    Tajjeeddine Rachidi

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Macario-Rat, G., Patarin, J. (2018). Two-Face: New Public Key Multivariate Schemes. In: Joux, A., Nitaj, A., Rachidi, T. (eds) Progress in Cryptology – AFRICACRYPT 2018. AFRICACRYPT 2018. Lecture Notes in Computer Science(), vol 10831. Springer, Cham. https://doi.org/10.1007/978-3-319-89339-6_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-89339-6_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-89338-9

  • Online ISBN: 978-3-319-89339-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation