Reassessing Security of Randomizable Signatures

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10808)

Abstract

The Camenisch-Lysyanskaya (CL) signature is a very popular tool in cryptography, especially among privacy-preserving constructions. Indeed, the latter benefit from their numerous features such as randomizability. Following the evolution of pairing-based cryptography, with the move from symmetric pairings to asymmetric pairings, Pointcheval and Sanders (PS) proposed at CT-RSA ’16 an alternative scheme which improves performances while keeping the same properties. Unfortunately, CL and PS signatures raise concerns in the cryptographic community because they both rely on interactive assumptions that essentially state their EUF-CMA security. This lack of precise security assessment is obviously a barrier to a widespread use of these signatures and a reason for preferring other constructions, such as the ones relying on q-type assumptions.

In this paper, we study more thoroughly the security of these signatures and prove that it actually relies, for both constructions, on simple variants of the \(\textsf {SDH}\) assumption, assuming a slight modification of the original constructions. Our work thus shows that the CL and PS signature schemes offer similar security guarantees as those provided by several other constructions using bilinear groups, and so that one can benefit from their interesting features without jeopardizing security.

Notes

Acknowledgments

This work was supported in part by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 339563 – CryptoCloud) and by the French ANR Project ANR-16-CE39-0014 PERSOCLOUD.

References

  1. 1.
    Abe, M., Groth, J., Haralambiev, K., Ohkubo, M.: Optimal structure-preserving signatures in asymmetric bilinear groups. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 649–666. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22792-9_37 CrossRefGoogle Scholar
  2. 2.
    Au, M.H., Susilo, W., Mu, Y.: Constant-size dynamic k-TAA. In: De Prisco, R., Yung, M. (eds.) SCN 2006. LNCS, vol. 4116, pp. 111–125. Springer, Heidelberg (2006).  https://doi.org/10.1007/11832072_8 CrossRefGoogle Scholar
  3. 3.
    Barić, N., Pfitzmann, B.: Collision-free accumulators and fail-stop signature schemes without trees. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 480–494. Springer, Heidelberg (1997).  https://doi.org/10.1007/3-540-69053-0_33 Google Scholar
  4. 4.
    Bellare, M., Micciancio, D., Warinschi, B.: Foundations of group signatures: formal definitions, simplified requirements, and a construction based on general assumptions. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 614–629. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-39200-9_38 CrossRefGoogle Scholar
  5. 5.
    Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: Ashby, V. (ed.) ACM CCS 1993, pp. 62–73. ACM Press, New York (1993)Google Scholar
  6. 6.
    Bernhard, D., Fuchsbauer, G., Ghadafi, E., Smart, N.P., Warinschi, B.: Anonymous attestation with user-controlled linkability. Int. J. Inf. Secur. 12(3), 219–249 (2013)CrossRefGoogle Scholar
  7. 7.
    Bichsel, P., Camenisch, J., Neven, G., Smart, N.P., Warinschi, B.: Get shorty via group signatures without encryption. In: Garay, J.A., De Prisco, R. (eds.) SCN 2010. LNCS, vol. 6280, pp. 381–398. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15317-4_24 CrossRefGoogle Scholar
  8. 8.
    Boneh, D., Boyen, X.: Short signatures without random oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 56–73. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24676-3_4 CrossRefGoogle Scholar
  9. 9.
    Boneh, D., Boyen, X.: Short signatures without random oracles and the SDH assumption in bilinear groups. J. Cryptol. 21(2), 149–177 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Brickell, E.F., Camenisch, J., Chen, L.: Direct anonymous attestation. In: Atluri, V., Pfitzmann, B., McDaniel, P. (eds.) ACM CCS 2004, pp. 132–145. ACM Press, New York (2004)Google Scholar
  11. 11.
    Camenisch, J., Lysyanskaya, A.: A signature scheme with efficient protocols. In: Cimato, S., Persiano, G., Galdi, C. (eds.) SCN 2002. LNCS, vol. 2576, pp. 268–289. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-36413-7_20 CrossRefGoogle Scholar
  12. 12.
    Camenisch, J., Lysyanskaya, A.: Signature schemes and anonymous credentials from bilinear maps. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 56–72. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-28628-8_4 CrossRefGoogle Scholar
  13. 13.
    Canard, S., Pointcheval, D., Sanders, O., Traoré, J.: Divisible e-cash made practical. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 77–100. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46447-2_4 Google Scholar
  14. 14.
    Chaum, D.: Blind signatures for untraceable payments. In: Chaum, D., Rivest, R.L., Sherman, A.T. (eds.) CRYPTO 1982, pp. 199–203. Plenum Press, New York (1982).  https://doi.org/10.1007/978-1-4757-0602-4_18 Google Scholar
  15. 15.
    Cramer, R., Shoup, V.: Signature schemes based on the strong RSA assumption. In: ACM CCS 1999, pp. 46–51. ACM Press, November 1999Google Scholar
  16. 16.
    Desmoulins, N., Lescuyer, R., Sanders, O., Traoré, J.: Direct anonymous attestations with dependent basename opening. In: Gritzalis, D., Kiayias, A., Askoxylakis, I. (eds.) CANS 2014. LNCS, vol. 8813, pp. 206–221. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-12280-9_14 Google Scholar
  17. 17.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theory 22(6), 644–654 (1976)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Freeman, D.M.: Converting pairing-based cryptosystems from composite-order groups to prime-order groups. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 44–61. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13190-5_3 CrossRefGoogle Scholar
  19. 19.
    Fujisaki, E., Okamoto, T.: Statistical zero knowledge protocols to prove modular polynomial relations. In: Kaliski, B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 16–30. Springer, Heidelberg (1997).  https://doi.org/10.1007/BFb0052225 CrossRefGoogle Scholar
  20. 20.
    Galbraith, S.D., Paterson, K.G., Smart, N.P.: Pairings for cryptographers. Discret. Appl. Math. 156(16), 3113–3121 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gerbush, M., Lewko, A., O’Neill, A., Waters, B.: Dual form signatures: an approach for proving security from static assumptions. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 25–42. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-34961-4_4 CrossRefGoogle Scholar
  22. 22.
    Ghadafi, E.: Short structure-preserving signatures. In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 305–321. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29485-8_18 CrossRefGoogle Scholar
  23. 23.
    Goldwasser, S., Micali, S., Rivest, R.L.: A digital signature scheme secure against adaptive chosen-message attacks. SIAM J. Comput. 17(2), 281–308 (1988)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Groth, J.: Efficient fully structure-preserving signatures for large messages. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015, Part I. LNCS, vol. 9452, pp. 239–259. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48797-6_11 CrossRefGoogle Scholar
  25. 25.
    Groth, J., Sahai, A.: Efficient non-interactive proof systems for bilinear groups. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 415–432. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78967-3_24 CrossRefGoogle Scholar
  26. 26.
    Guillevic, A.: Comparing the pairing efficiency over composite-order and prime-order elliptic curves. In: Jacobson, M., Locasto, M., Mohassel, P., Safavi-Naini, R. (eds.) ACNS 2013. LNCS, vol. 7954, pp. 357–372. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38980-1_22 CrossRefGoogle Scholar
  27. 27.
    Krawczyk, H., Rabin, T.: Chameleon signatures. In: NDSS 2000. The Internet Society, February 2000Google Scholar
  28. 28.
    Lee, K., Lee, D.H., Yung, M.: Aggregating CL-signatures revisited: extended functionality and better efficiency. In: Sadeghi, A.-R. (ed.) FC 2013. LNCS, vol. 7859, pp. 171–188. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39884-1_14 CrossRefGoogle Scholar
  29. 29.
    Libert, B., Mouhartem, F., Peters, T., Yung, M.: Practical “signatures with efficient protocols” from simple assumptions. In: Chen, X., Wang, X., Huang, X. (eds.) ASIACCS 2016, pp. 511–522. ACM Press, New York (2016)Google Scholar
  30. 30.
    Okamoto, T.: Efficient blind and partially blind signatures without random oracles. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 80–99. Springer, Heidelberg (2006).  https://doi.org/10.1007/11681878_5 CrossRefGoogle Scholar
  31. 31.
    Pointcheval, D., Sanders, O.: Short randomizable signatures. In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 111–126. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29485-8_7 CrossRefGoogle Scholar
  32. 32.
    Pointcheval, D., Sanders, O.: Reassessing security of randomizable signatures (full version). Cryptology ePrint Archive, Report 2017/1197 (2017)Google Scholar
  33. 33.
    Schäge, S.: Tight security for signature schemes without random oracles. J. Cryptol. 28(3), 641–670 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.DIENS, École normale supérieure, CNRSPSL Research UniversityParisFrance
  2. 2.INRIAParisFrance
  3. 3.Orange LabsApplied Crypto GroupCesson-SévignéFrance

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