Skip to main content
Download book PDF
View expanded cover

International Conference on the Theory and Application of Cryptology and Information Security

ASIACRYPT 2014: Advances in Cryptology – ASIACRYPT 2014 pp 366–385Cite as

Black-Box Separations for One-More (Static) CDH and Its Generalization

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNSC,volume 8874)

Abstract

As one-more problems are widely used in both proving and analyzing the security of various cryptographic schemes, it is of fundamental importance to investigate the hardness of the one-more problems themselves. Bresson et al. (CT-RSA ’08) first showed that it is difficult to rely the hardness of some one-more problems on the hardness of their “regular” ones. Pass (STOC ’11) then gave a stronger black-box separation showing that the hardness of some one-more problems cannot be based on standard assumptions using black-box reductions. However, since previous works only deal with one-more problems whose solution can be efficiently checked, the relation between the hardness of the one-more (static) CDH problem over non-bilinear groups and other hard problems is still unclear. In this work, we give the first impossibility results showing that black-box reductions cannot be used to base the hardness of the one-more (static) CDH problem (over groups where the DDH problem is still hard) on any standard hardness assumption. Furthermore, we also extend the impossibility results to a class of generalized “one-more” problems, which not only subsume/strengthen many existing separations for traditional one-more problems, but also give new separations for many other interesting “one-more” problems.

Keywords

  • Blind Signature
  • Test Algorithm
  • Security Parameter
  • Discrete Logarithm Problem
  • Impossibility Result

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Jiang Zhang, Zhenfeng Zhang and Yanfei Guo are sponsored by the National Basic Research Program of China under Grant No. 2013CB338003, and the National Natural Science Foundation of China (NSFC) under Grant No. 61170278, 91118006. Yu Chen is sponsored by NSFC under Grant No. 61303257, and the Strategic Priority Research Program of CAS under Grant No. XDA06010701. Zongyang Zhang is an International Research Fellow of JSPS and his work is in part supported by NSFC under grant No. 61303201.

Download conference paper PDF

References

  1. Abadi, M., Feigenbaum, J., Kilian, J.: On hiding information from an oracle. Journal of Computer and System Sciences 39(1), 21–50 (1989)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Bellare, M., Namprempre, C., Neven, G.: Security proofs for identity-based identification and signature schemes. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 268–286. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  3. Bellare, M., Namprempre, C., Neven, G.: Security proofs for identity-based identification and signature schemes. Journal of Cryptology 22(1), 1–61 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Bellare, M., Namprempre, C., Pointcheval, D., Semanko, M.: The power of RSA inversion oracles and the security of Chaum’s RSA-based blind signature scheme. In: Syverson, P.F. (ed.) FC 2001. LNCS, vol. 2339, pp. 309–328. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  5. Bellare, M., Namprempre, C., Pointcheval, D., Semanko, M.: The one-more-RSA-inversion problems and the security of Chaum’s blind signature scheme. Journal of Cryptology 16(3), 185–215 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Bellare, M., Neven, G.: Transitive signatures: new schemes and proofs. IEEE Transactions on Information Theory 51(6), 2133–2151 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Bellare, M., Palacio, A.: GQ and Schnorr identification schemes: Proofs of security against impersonation under active and concurrent attacks. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 162–177. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  8. Boldyreva, A.: Threshold signatures, multisignatures and blind signatures based on the Gap-Diffie-Hellman-group signature scheme. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 31–46. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  9. Boneh, D., Venkatesan, R.: Breaking RSA may not be equivalent to factoring. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 59–71. Springer, Heidelberg (1998)

    CrossRef  Google Scholar 

  10. Bresson, E., Monnerat, J., Vergnaud, D.: Separation results on the “one-more” computational problems. In: Malkin, T. (ed.) CT-RSA 2008. LNCS, vol. 4964, pp. 71–87. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  11. Brown, D.R.L.: Irreducibility to the one-more evaluation problems: More may be less. Cryptology ePrint Archive, Report 2007/435 (2007)

    Google Scholar 

  12. Brown, D.R.L., Gallant, R.P.: The static Diffie-Hellman problem. Cryptology ePrint Archive, Report 2004/306 (2004)

    Google Scholar 

  13. Canard, S., Gouget, A., Traoré, J.: Improvement of efficiency in (unconditional) anonymous transferable e-cash. In: Tsudik, G. (ed.) FC 2008. LNCS, vol. 5143, pp. 202–214. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  14. Canetti, R., Gennaro, R., Jarecki, S., Krawczyk, H., Rabin, T.: Adaptive security for threshold cryptosystems. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 98–115. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  15. Canetti, R., Lin, H., Pass, R.: Adaptive hardness and composable security in the plain model from standard assumptions. In: FOCS, pp. 541–550 (2010)

    Google Scholar 

  16. Cash, D., Kiltz, E., Shoup, V.: The twin Diffie-Hellman problem and applications. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 127–145. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  17. Chaum, D.: Blind signatures for untraceable payments. In: CRYPTO, pp. 199–203 (1982)

    Google Scholar 

  18. Chen, Y., Huang, Q., Zhang, Z.: Sakai-ohgishi-kasahara identity-based non-interactive key exchange scheme, revisited. In: Susilo, W., Mu, Y. (eds.) ACISP 2014. LNCS, vol. 8544, pp. 274–289. Springer, Heidelberg (2014)

    CrossRef  Google Scholar 

  19. De Cristofaro, E., Tsudik, G.: Practical private set intersection protocols with linear complexity. In: Sion, R. (ed.) FC 2010. LNCS, vol. 6052, pp. 143–159. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  20. Deng, Y., Goyal, V., Sahai, A.: Resolving the simultaneous resettability conjecture and a new non-black-box simulation strategy. In: FOCS, pp. 251–260 (2009)

    Google Scholar 

  21. Dodis, Y., Haitner, I., Tentes, A.: On the instantiability of hash-and-sign RSA signatures. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 112–132. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  22. Dwork, C., Naor, M., Sahai, A.: Concurrent zero-knowledge. Journal of the ACM 51(6), 851–898 (2004)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Fiore, D., Schröder, D.: Uniqueness is a different story: Impossibility of verifiable random functions from trapdoor permutations. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 636–653. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  24. Fischlin, M.: Black-box reductions and separations in cryptography. In: Mitrokotsa, A., Vaudenay, S. (eds.) AFRICACRYPT 2012. LNCS, vol. 7374, pp. 413–422. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  25. Fischlin, M., Fleischhacker, N.: Limitations of the meta-reduction technique: The case of Schnorr signatures. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 444–460. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  26. Fischlin, M., Schröder, D.: On the impossibility of three-move blind signature schemes. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 197–215. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  27. Garg, S., Bhaskar, R., Lokam, S.V.: Improved bounds on security reductions for discrete log based signatures. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 93–107. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  28. Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: STOC, pp. 99–108 (2011)

    Google Scholar 

  29. Granger, R.: On the static Diffie-Hellman problem on elliptic curves over extension fields. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 283–302. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  30. Herranz, J., Laguillaumie, F.: Blind ring signatures secure under the chosen-target-CDH assumption. In: Katsikas, S.K., López, J., Backes, M., Gritzalis, S., Preneel, B. (eds.) ISC 2006. LNCS, vol. 4176, pp. 117–130. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  31. Joux, A., Lercier, R., Naccache, D., Thomé, E.: Oracle-assisted static Diffie-Hellman is easier than discrete logarithms. In: Parker, M.G. (ed.) Cryptography and Coding 2009. LNCS, vol. 5921, pp. 351–367. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  32. Juels, A., Luby, M., Ostrovsky, R.: Security of blind digital signatures. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 150–164. Springer, Heidelberg (1997)

    CrossRef  Google Scholar 

  33. Katz, J., Schröder, D., Yerukhimovich, A.: Impossibility of blind signatures from one-way permutations. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 615–629. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  34. Koblitz, N., Menezes, A.: Another look at non-standard discrete log and Diffie-Hellman problems. Cryptology ePrint Archive, Report 2007/442 (2007)

    Google Scholar 

  35. Okamoto, T., Pointcheval, D.: The Gap-problems: A new class of problems for the security of cryptographic schemes. In: Kim, K.-C. (ed.) PKC 2001. LNCS, vol. 1992, pp. 104–118. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  36. Paillier, P., Vergnaud, D.: Discrete-log-based signatures may not be equivalent to discrete log. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 1–20. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  37. Pass, R.: Limits of provable security from standard assumptions. In: STOC, pp. 109–118 (2011)

    Google Scholar 

  38. Pass, R., Venkitasubramaniam, M.: On constant-round concurrent zero-knowledge. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 553–570. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  39. Pointcheval, D., Stern, J.: Security arguments for digital signatures and blind signatures. Journal of Cryptology 13(3), 361–396 (2000)

    CrossRef  MATH  Google Scholar 

  40. Prabhakaran, M., Rosen, A., Sahai, A.: Concurrent zero knowledge with logarithmic round-complexity. In: FOCS, pp. 366–375 (2002)

    Google Scholar 

  41. Richardson, R., Kilian, J.: On the concurrent composition of zero-knowledge proofs. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 415–431. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  42. Seurin, Y.: On the exact security of Schnorr-type signatures in the random oracle model. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 554–571. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Trusted Computing and Information Assurance Laboratory, State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, P.R. China

    Jiang Zhang, Zhenfeng Zhang & Yanfei Guo

  2. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, P.R. China

    Yu Chen

  3. National Institute of Advanced Industrial Science and Technology (AIST), Japan

    Zongyang Zhang

Authors
  1. Jiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenfeng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yanfei Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zongyang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Applied Statistics Unit, Indian Statistical Institute, 203, B.T.Road, 700108, Kolkata,, India

    Palash Sarkar

  2. Nagoya University, Japan

    Tetsu Iwata

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 International Association for Cryptologic Research

About this paper

Cite this paper

Zhang, J., Zhang, Z., Chen, Y., Guo, Y., Zhang, Z. (2014). Black-Box Separations for One-More (Static) CDH and Its Generalization. In: Sarkar, P., Iwata, T. (eds) Advances in Cryptology – ASIACRYPT 2014. ASIACRYPT 2014. Lecture Notes in Computer Science, vol 8874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45608-8_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-45608-8_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-45607-1

  • Online ISBN: 978-3-662-45608-8

  • eBook Packages: Computer ScienceComputer Science (R0)