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Cryptanalysis of the Multivariate Encryption Scheme EFLASH

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Topics in Cryptology – CT-RSA 2020 (CT-RSA 2020)

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Abstract

EFLASH is a multivariate public-key encryption scheme proposed by Cartor and Smith-Tone at SAC 2018. In this paper we investigate the hardness of solving the particular equation systems arising from EFLASH, and show that the solving degree for these types of systems is much lower than estimated by the authors. We show that a Gröbner basis algorithm will produce degree fall polynomials at a low degree for EFLASH systems. In particular we are able to accurately predict the number of these polynomials occurring at step degrees 3 and 4 in our attacks. We performed several experiments using the computer algebra system MAGMA, which indicate that the solving degree is at most one higher than the one where degree fall polynomials occur; moreover, our experiments show that whenever the predicted number of degree fall polynomials is positive, it is exact. Our conclusion is that EFLASH does not offer the level of security claimed by the designers. In particular, we estimate that the EFLASH version with 80-bit security parameters offers at most 69 bits of security.

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Notes

  1. 1.

    The authors call this the degree of regularity, but are in fact describing the first fall degree.

References

  1. Bardet, M., Faugère, J.-C., Salvy, B.: Complexity of Gröbner basis computation for Semi-regular overdetermined sequences over \({{\mathbb{F}}_{2}}\) with solutions in \({{\mathbb{F}}_{2}}\). [Research Report] RR-5049, INRIA, inria-00071534 (2003)

    Google Scholar 

  2. Bardet, M., et al.: Asymptotic behaviour of the degree of regularity of semi-regular polynomial systems. In: Proceedings of MEGA, vol. 5 (2005)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Cabarcas, D., Smith-Tone, D., Verbel, J.A.: Key recovery attack for ZHFE. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 289–308. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59879-6_17

    Chapter  Google Scholar 

  6. Cartor, R., Smith-Tone, D.: EFLASH: a new multivariate encryption scheme. In: Cid, C., Jacobson Jr., M. (eds.) SAC 2018. Lecture Notes in Computer Science, vol. 11349, pp. 281–299. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10970-7_13

    Chapter  Google Scholar 

  7. Chen, M.-S., Yang, B.-Y., Smith-Tone, D.: PFLASH - secure asymmetric signatures on smart cards. In: Lightweight Cryptography Workshop (2015). https://ws680.nist.gov/publication/get_pdf.cfm?pub_id=926103

  8. Cox, D.A., Little, J., O’shea, D.: Using Algebraic Geometry, vol. 185. Springer, New York (2006). https://doi.org/10.1007/b138611

    Book  MATH  Google Scholar 

  9. Diene, A., Ding, J., Gower, J.E., Hodges, T.J., Yin, Z.: Dimension of the linearization equations of the matsumoto-imai cryptosystems. In: Ytrehus, Ø. (ed.) WCC 2005. LNCS, vol. 3969, pp. 242–251. Springer, Heidelberg (2006). https://doi.org/10.1007/11779360_20

    Chapter  Google Scholar 

  10. 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). https://doi.org/10.1007/978-3-642-22792-9_41

    Chapter  Google Scholar 

  11. Ding, J., Kleinjung, T.: Degree of regularity for HFE-. In: IACR Cryptology ePrint Archive 2011, p. 570 (2011)

    Google Scholar 

  12. Ding, J., Schmidt, D.: Solving degree and degree of regularity for polynomial systems over a finite fields. In: Fischlin, M., Katzenbeisser, S. (eds.) Number Theory and Cryptography. LNCS, vol. 8260, pp. 34–49. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42001-6_4

    Chapter  Google Scholar 

  13. Ding, J., Dubois, V., Yang, B.-Y., Chen, O.C.-H., Cheng, C.-M.: Could SFLASH be repaired? In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5126, pp. 691–701. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70583-3_56

    Chapter  Google Scholar 

  14. Faugère, J.C.: A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)

    Article  MathSciNet  Google Scholar 

  15. Faugère, J.C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pp. 75–83. ACM (2002)

    Google Scholar 

  16. 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 

  17. Felke, P.: On the affine transformations of HFE-cryptosystems and systems with branches. In: Ytrehus, Ø. (ed.) WCC 2005. LNCS, vol. 3969, pp. 229–241. Springer, Heidelberg (2006). https://doi.org/10.1007/11779360_19

    Chapter  MATH  Google Scholar 

  18. Felke, P.: On the security of biquadratic \(C^{*}\) public-key cryptosystems and its generalizations. Crypt. Commun. 11, 1–16 (2018)

    MathSciNet  Google Scholar 

  19. Joux, A., Vitse, V.: A crossbred algorithm for solving boolean polynomial systems. In: Kaczorowski, J., Pieprzyk, J., Pomykała, J. (eds.) NuTMiC 2017. LNCS, vol. 10737, pp. 3–21. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76620-1_1

    Chapter  Google Scholar 

  20. 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). https://doi.org/10.1007/3-540-48405-1_2

    Chapter  Google Scholar 

  21. Liu, J., et al.: Structural key recovery of simple matrix encryption scheme family. Comput. J. 61 (2018). https://doi.org/10.1093/comjnl/bxy093

  22. Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In: Barstow, D., et al. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988). https://doi.org/10.1007/3-540-45961-8_39

    Chapter  Google Scholar 

  23. Patarin, J.: Cryptanalysis of the matsumoto and imai public key scheme of Eurocrypt’88. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-44750-4_20

    Chapter  Google Scholar 

  24. Patarin, J.: Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of asymmetric algorithms. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68339-9_4

    Chapter  Google Scholar 

  25. 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). https://doi.org/10.1007/3-540-45353-9_22

    Chapter  Google Scholar 

  26. Perlner, R., Petzoldt, A., Smith-Tone, D.: Total break of the SRP encryption scheme. In: Adams, C., Camenisch, J. (eds.) SAC 2017. LNCS, vol. 10719, pp. 355–373. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72565-9_18

    Chapter  Google Scholar 

  27. 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, Cham (2014). https://doi.org/10.1007/978-3-319-11659-4_14

    Chapter  Google Scholar 

  28. 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). https://doi.org/10.1007/978-3-642-38616-9_16

    Chapter  Google Scholar 

  29. Yasuda, T., Sakurai, K.: A multivariate encryption scheme with rainbow. In: Qing, S., Okamoto, E., Kim, K., Liu, D. (eds.) ICICS 2015. LNCS, vol. 9543, pp. 236–251. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29814-6_19

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Simula UiB, Bergen, Norway

    Morten Øygarden, Håvard Raddum & Carlos Cid

  2. University of Applied Sciences Emden-Leer, Emden, Germany

    Patrick Felke

  3. Royal Holloway University of London, Egham, UK

    Carlos Cid

Authors
  1. Morten Øygarden
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  2. Patrick Felke
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  3. Håvard Raddum
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  4. Carlos Cid
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Correspondence to Morten Øygarden .

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Editors and Affiliations

  1. University of California, Irvine, CA, USA

    Stanislaw Jarecki

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Øygarden, M., Felke, P., Raddum, H., Cid, C. (2020). Cryptanalysis of the Multivariate Encryption Scheme EFLASH. In: Jarecki, S. (eds) Topics in Cryptology – CT-RSA 2020. CT-RSA 2020. Lecture Notes in Computer Science(), vol 12006. Springer, Cham. https://doi.org/10.1007/978-3-030-40186-3_5

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  • DOI: https://doi.org/10.1007/978-3-030-40186-3_5

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  • Online ISBN: 978-3-030-40186-3

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