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Wave: A New Family of Trapdoor One-Way Preimage Sampleable Functions Based on Codes

  • Thomas Debris-AlazardEmail author
  • Nicolas Sendrier
  • Jean-Pierre Tillich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11921)

Abstract

We present here a new family of trapdoor one-way functions that are Preimage Sampleable on Average (PSA) based on codes, the Wave-PSA family. The trapdoor function is one-way under two computational assumptions: the hardness of generic decoding for high weights and the indistinguishability of generalized \((U,U+V)\)-codes. Our proof follows the GPV strategy [28]. By including rejection sampling, we ensure the proper distribution for the trapdoor inverse output. The domain sampling property of our family is ensured by using and proving a variant of the left-over hash lemma. We instantiate the new Wave-PSA family with ternary generalized \((U,U+V)\)-codes to design a “hash-and-sign” signature scheme which achieves existential unforgeability under adaptive chosen message attacks (EUF-CMA) in the random oracle model.

Notes

Acknowledgements

We wish to thank the anonymous reviewers. In particular, our warmest gratitude goes to the last of them whose work went much beyond what can be found in a standard review. This includes the link clarifying our definition of “preimage sampleable on average” with the GPV definition [28] given in Sect. 3.1, a reorganization of the paper focusing on the main theoretical contribution, and simplifications and/or clarifications that all helped a great deal to improve this paper. We are also indebted to André Chailloux, Léo Ducas and Thomas Prest for their early interest, insightful suggestions, and unwavering support.

References

  1. 1.
    Alkim, E., et al.: Revisiting TESLA in the quantum random oracle model. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 143–162. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-59879-6_9CrossRefGoogle Scholar
  2. 2.
    Aragon, N., Blazy, O., Gaborit, P., Hauteville, A., Zémor, G.: Durandal: a rank metric based signature scheme. IACR Cryptology ePrint Archive (2018), Report 2018/1192, December 2018Google Scholar
  3. 3.
    Bader, C., Jager, T., Li, Y., Schäge, S.: On the impossibility of tight cryptographic reductions. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 273–304. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_10CrossRefGoogle Scholar
  4. 4.
    Baldi, M., Bianchi, M., Chiaraluce, F., Rosenthal, J., Schipani, D.: Using LDGM codes and sparse syndromes to achieve digital signatures. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 1–15. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38616-9_1CrossRefGoogle Scholar
  5. 5.
    Barak, B., et al.: Leftover hash lemma, revisited. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 1–20. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22792-9_1CrossRefGoogle Scholar
  6. 6.
    Barg, A.: Complexity issues in coding theory. Electronic Colloquium on Computational Complexity, October 1997. https://eccc.weizmann.ac.il/eccc-reports/1997/TR97-046/Paper.pdf
  7. 7.
    Barreto, P.S., Misoczki, R., Simplicio, M.A.J.: One-time signature scheme from syndrome decoding over generic error-correcting codes. J. Syst. Softw. 84(2), 198–204 (2011)CrossRefGoogle Scholar
  8. 8.
    Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in 2n/20: How \(1+1=0\) improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 520–536. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29011-4_31CrossRefGoogle Scholar
  9. 9.
    Bellare, M., Rogaway, P.: The exact security of digital signatures-how to sign with RSA and Rabin. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 399–416. Springer, Heidelberg (1996).  https://doi.org/10.1007/3-540-68339-9_34CrossRefGoogle Scholar
  10. 10.
    Bernstein, D.J., Chou, T., Schwabe, P.: McBits: fast constant-time code-based cryptography. In: Bertoni, G., Coron, J.-S. (eds.) CHES 2013. LNCS, vol. 8086, pp. 250–272. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40349-1_15CrossRefGoogle Scholar
  11. 11.
    Both, L., May, A.: Decoding linear codes with high error rate and its impact for LPN security. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 25–46. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-79063-3_2CrossRefGoogle Scholar
  12. 12.
    Bricout, R., Chailloux, A., Debris-Alazard, T., Lequesne, M.: Ternary syndrome decoding with large weights. preprint, arXiv:1903.07464, February 2019. To appear in the proceedings of SAC 2019
  13. 13.
    Cayrel, P.-L., Otmani, A., Vergnaud, D.: On Kabatianskii-Krouk-Smeets signatures. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 237–251. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-73074-3_18CrossRefGoogle Scholar
  14. 14.
    Coron, J.-S.: Optimal security proofs for PSS and other signature schemes. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 272–287. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-46035-7_18CrossRefGoogle Scholar
  15. 15.
    Courtois, N.T., Finiasz, M., Sendrier, N.: How to achieve a McEliece-based digital signature scheme. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 157–174. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45682-1_10CrossRefGoogle Scholar
  16. 16.
    Debris-Alazard, T., Sendrier, N., Tillich, J.P.: A new signature scheme based on \((U|U+V)\) codes. preprint, arXiv:1706.08065v1, June 2017
  17. 17.
    Debris-Alazard, T., Sendrier, N., Tillich, J.P.: The problem with the surf scheme. preprint, arXiv:1706.08065, November 2017
  18. 18.
    Debris-Alazard, T., Sendrier, N., Tillich, J.P.: Wave: A new family of trapdoor one-way preimage sampleable functions based on codes. Cryptology ePrint Archive, Report 2018/996, May 2019. Full version of the current paper. All statement and section numbers quoted in this paper refer specifically to the May 2019 versionGoogle Scholar
  19. 19.
    Debris-Alazard, T., Tillich, J.P.: Statistical decoding. preprint, arXiv:1701.07416, January 2017
  20. 20.
    Debris-Alazard, T., Tillich, J.-P.: Two attacks on rank metric code-based schemes: ranksign and an IBE scheme. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018. LNCS, vol. 11272, pp. 62–92. Springer, Cham (2018).  https://doi.org/10.1007/978-3-030-03326-2_3CrossRefGoogle Scholar
  21. 21.
    Dumer, I.: On minimum distance decoding of linear codes. In: Proceedings of 5th Joint Soviet-Swedish International Workshop Information Theory, pp. 50–52. Moscow (1991)Google Scholar
  22. 22.
    Faugère, J.C., Gauthier, V., Otmani, A., Perret, L., Tillich, J.P.: A distinguisher for high rate McEliece cryptosystems. In: Proceedings of IEEE Information Theory Workshop- ITW 2011, pp. 282–286. Paraty, Brasil, October 2011Google Scholar
  23. 23.
    Finiasz, M.: Parallel-CFS- strengthening the CFS McEliece-based signature scheme. In: Biryukov, A., Gong, G., Stinson, D.R. (eds.) SAC 2010. LNCS, vol. 6544, pp. 159–170. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19574-7_11CrossRefGoogle Scholar
  24. 24.
    Fouque, P.A., et al.: Falcon: fast-fourier lattice-based compact signatures over NTRUGoogle Scholar
  25. 25.
    Fukushima, K., Roy, P.S., Xu, R., Kiyomoto, S., Morozov, K., Takagi, T.: RaCoSS (random code-based signature scheme). first round submission to the NIST post-quantum cryptography call, November 2017Google Scholar
  26. 26.
    Gaborit, P., Ruatta, O., Schrek, J., Zémor, G.: New results for rank-based cryptography. In: Pointcheval, D., Vergnaud, D. (eds.) AFRICACRYPT 2014. LNCS, vol. 8469, pp. 1–12. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-06734-6_1CrossRefGoogle Scholar
  27. 27.
    Gaborit, P., Schrek, J.: Efficient code-based one-time signature from automorphism groups with syndrome compatibility. In: Proceedings IEEE International Symposium Information Theory - ISIT 2012, pp. 1982–1986. Cambridge, MA, USA, July 2012Google Scholar
  28. 28.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pp. 197–206. ACM (2008)Google Scholar
  29. 29.
    Gligoroski, D., Samardjiska, S., Jacobsen, H., Bezzateev, S.: McEliece in the world of Escher. IACR Cryptology ePrint Archive, Report 2014/360 (2014)Google Scholar
  30. 30.
    Goldwasser, S., Micciancio, D.: Complexity of lattice problems: a cryptographic perspective. In: Kluwer International Series in Engineering and Computer Science, vol. 671. Kluwer Academic Publishers, Dordrecht, March 2002Google Scholar
  31. 31.
    Huelsing, A., Bernstein, D.J., Panny, L., Lange, T.: Official NIST comments made for RaCoSS, official NIST comments made for RaCoSS (2018)Google Scholar
  32. 32.
    Johansson, T., Jönsson, F.: On the complexity of some cryptographic problems based on the general decoding problem. IEEE Trans. Inform. Theory 48(10), 2669–2678 (2002)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kabatianskii, G., Krouk, E., Semenov, S.: Error Correcting Coding and Security for Data Networks: Analysis of the Superchannel Concept. Wiley, Hoboken (2005)Google Scholar
  34. 34.
    Kabatianskii, G., Krouk, E., Smeets, B.: A digital signature scheme based on random error-correcting codes. In: Darnell, M. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 161–167. Springer, Heidelberg (1997).  https://doi.org/10.1007/BFb0024461CrossRefGoogle Scholar
  35. 35.
    Landais, G., Sendrier, N.: Implementing CFS. In: Galbraith, S., Nandi, M. (eds.) INDOCRYPT 2012. LNCS, vol. 7668, pp. 474–488. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-34931-7_27CrossRefGoogle Scholar
  36. 36.
    Lee, W., Kim, Y.S., Lee, Y.W., No, J.S.: Post quantum signature scheme based on modified Reed-Muller code pqsigRM. first round submission to the NIST postquantum cryptography call, November 2017Google Scholar
  37. 37.
    Lyubashevsky, V.: Fiat-shamir with aborts: applications to lattice and factoring-based signatures. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 598–616. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-10366-7_35 CrossRefGoogle Scholar
  38. 38.
    May, A., Meurer, A., Thomae, E.: Decoding random linear codes in \(\tilde{\cal{O}}(2^{0.054n})\). In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 107–124. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-25385-0_6CrossRefzbMATHGoogle Scholar
  39. 39.
    May, A., Ozerov, I.: On computing nearest neighbors with applications to decoding of binary linear codes. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 203–228. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46800-5_9CrossRefGoogle Scholar
  40. 40.
    Moody, D., Perlner, R.: Vulnerabilities of McEliece in the world of Escher. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 104–117. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29360-8_8CrossRefGoogle Scholar
  41. 41.
    Otmani, A., Tillich, J.-P.: An efficient attack on all concrete KKS proposals. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 98–116. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-25405-5_7CrossRefGoogle Scholar
  42. 42.
    Phesso, A., Tillich, J.-P.: An efficient attack on a code-based signature scheme. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 86–103. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29360-8_7CrossRefGoogle Scholar
  43. 43.
    Prange, E.: The use of information sets in decoding cyclic codes. IRE Trans. Inf. Theory 8(5), 5–9 (1962)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Sendrier, N.: Decoding one out of many. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 51–67. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-25405-5_4CrossRefGoogle Scholar
  45. 45.
    Stern, J.: A method for finding codewords of small weight. In: Cohen, G., Wolfmann, J. (eds.) Coding Theory 1988. LNCS, vol. 388, pp. 106–113. Springer, Heidelberg (1989).  https://doi.org/10.1007/BFb0019850CrossRefGoogle Scholar
  46. 46.
    Stern, J.: A new identification scheme based on syndrome decoding. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 13–21. Springer, Heidelberg (1994).  https://doi.org/10.1007/3-540-48329-2_2CrossRefGoogle Scholar
  47. 47.
    Wagner, D.: A generalized birthday problem. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 288–304. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45708-9_19CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  • Thomas Debris-Alazard
    • 1
    • 2
    Email author
  • Nicolas Sendrier
    • 2
  • Jean-Pierre Tillich
    • 2
  1. 1.Sorbonne Université, Collège DoctoralParisFrance
  2. 2.InriaParisFrance

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