Advertisement

Combiners for Backdoored Random Oracles

  • Balthazar Bauer
  • Pooya Farshim
  • Sogol Mazaheri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10992)

Abstract

We formulate and study the security of cryptographic hash functions in the backdoored random-oracle (BRO) model, whereby a big brother designs a “good” hash function, but can also see arbitrary functions of its table via backdoor capabilities. This model captures intentional (and unintentional) weaknesses due to the existence of collision-finding or inversion algorithms, but goes well beyond them by allowing, for example, to search for structured preimages. The latter can easily break constructions that are secure under random inversions.

BROs make the task of bootstrapping cryptographic hardness somewhat challenging. Indeed, with only a single arbitrarily backdoored function no hardness can be bootstrapped as any construction can be inverted. However, when two (or more) independent hash functions are available, hardness emerges even with unrestricted and adaptive access to all backdoor oracles. At the core of our results lie new reductions from cryptographic problems to the communication complexities of various two-party tasks. Along the way we establish a communication complexity lower bound for set-intersection for cryptographically relevant ranges of parameters and distributions and where set-disjointness can be easy.

Keywords

Random oracle Combiner Communication complexity Set-disjointness Set-intersection Lower bounds 

Notes

Acknowledgments

We thank Marc Fischlin for participating in the early stages of this work. We also thank the CRYPTO’18 (sub)reviewers for their valuable comments. Bauer was supported by the French ANR Project ANR-16-CE39-0002 EfTrEC. Farshim was supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 339563 - CryptoCloud). Mazaheri was supported by the German Federal Ministry of Education and Research (BMBF) and by the Hessian State Ministry for Higher Education, Research and the Arts, within CRISP.

References

  1. 1.
    Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory (preliminary version). In: 27th FOCS, pp. 337–347 (1986)Google Scholar
  2. 2.
    Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. In: 43rd FOCS, pp. 209–218 (2002)Google Scholar
  3. 3.
    Barak, B., Braverman, M., Chen, X., Rao, A.: How to compress interactive communication. In: 42nd ACM STOC, pp. 67–76 (2010)Google Scholar
  4. 4.
    Bauer, B., Farshim, P., Mazaheri, S.: Combiners for backdoored random oracles. Cryptology ePrint Archive (2018)Google Scholar
  5. 5.
    Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: ACM CCS 1993, pp. 62–73 (1993)Google Scholar
  6. 6.
    Bernstein, D.J., Lange, T., Niederhagen, R.: Dual EC: a standardized back door. Cryptology ePrint Archive, Report 2015/767 (2015). http://eprint.iacr.org/2015/767
  7. 7.
    Boneh, D., Boyen, X.: On the impossibility of efficiently combining collision resistant hash functions. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 570–583. Springer, Heidelberg (2006).  https://doi.org/10.1007/11818175_34CrossRefGoogle Scholar
  8. 8.
    Canetti, R., Rivest, R.L., Sudan, M., Trevisan, L., Vadhan, S.P., Wee, H.: Amplifying collision resistance: a complexity-theoretic treatment. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 264–283. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74143-5_15CrossRefGoogle Scholar
  9. 9.
    Chattopadhyay, A., Pitassi, T.: The story of set disjointness. SIGACT News 41(3), 59–85 (2010)CrossRefGoogle Scholar
  10. 10.
    Checkoway, S., Maskiewicz, J., Garman, C., Fried, J., Cohney, S., Green, M., Heninger, N., Weinmann, R.-P., Rescorla, E., Shacham, H.: A systematic analysis of the juniper dual EC incident. In: ACM CCS 2016, pp. 468–479 (2016)Google Scholar
  11. 11.
    Checkoway, S., et al.: On the practical exploitability of dual EC in TLS implementations. In: 23rd USENIX Security Symposium (USENIX Security 14), pp. 319–335 (2014)Google Scholar
  12. 12.
    Coretti, S., Dodis, Y., Guo, S., Steinberger, J.: Random oracles and non-uniformity. Cryptology ePrint Archive, Report 2017/937 (2017). http://eprint.iacr.org/2017/937
  13. 13.
    Dinur, I.: New attacks on the concatenation and XOR hash combiners. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part I. LNCS, vol. 9665, pp. 484–508. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49890-3_19CrossRefGoogle Scholar
  14. 14.
    Dodis, Y., Guo, S., Katz, J.: Fixing cracks in the concrete: random oracles with auxiliary input, revisited. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part II. LNCS, vol. 10211, pp. 473–495. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56614-6_16CrossRefGoogle Scholar
  15. 15.
    Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987).  https://doi.org/10.1007/3-540-47721-7_12CrossRefGoogle Scholar
  16. 16.
    Fischlin, M., Lehmann, A.: Security-amplifying combiners for collision-resistant hash functions. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 224–243. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74143-5_13CrossRefGoogle Scholar
  17. 17.
    Fischlin, M., Lehmann, A., Pietrzak, K.: Robust multi-property combiners for hash functions. J. Cryptol. 27(3), 397–428 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Goldreich, O.: Foundations of Cryptography: Basic Tools, vol. 1. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  19. 19.
    Guruswami, V., Cheraghchi, M.: Set disjointness lower bound via product distribution. Scribes for Information theory and its applications in theory of computation (2013). http://www.cs.cmu.edu/~venkatg/teaching/ITCS-spr2013/
  20. 20.
    Hoch, J.J., Shamir, A.: On the strength of the concatenated hash combiner when all the hash functions are weak. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 616–630. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-70583-3_50CrossRefGoogle Scholar
  21. 21.
    Joux, A.: Multicollisions in iterated hash functions. Application to cascaded constructions. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 306–316. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-28628-8_19CrossRefzbMATHGoogle Scholar
  22. 22.
    Katz, J., Lucks, S., Thiruvengadam, A.: Hash functions from defective ideal ciphers. In: Nyberg, K. (ed.) CT-RSA 2015. LNCS, vol. 9048, pp. 273–290. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-16715-2_15CrossRefGoogle Scholar
  23. 23.
    Kawachi, A., Numayama, A., Tanaka, K., Xagawa, K.: Security of encryption schemes in weakened random oracle models. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 403–419. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13013-7_24CrossRefGoogle Scholar
  24. 24.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, New York (1997)CrossRefGoogle Scholar
  25. 25.
    Lehmann, A.: On the security of hash function combiners. Ph.D. thesis, TU Darmstadt (2010)Google Scholar
  26. 26.
    Leurent, G., Wang, L.: The sum can be weaker than each part. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part I. LNCS, vol. 9056, pp. 345–367. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46800-5_14CrossRefGoogle Scholar
  27. 27.
    Liskov, M.: Constructing an ideal hash function from weak ideal compression functions. In: Biham, E., Youssef, A.M. (eds.) SAC 2006. LNCS, vol. 4356, pp. 358–375. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74462-7_25CrossRefGoogle Scholar
  28. 28.
    Lucks, S.: A failure-friendly design principle for hash functions. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 474–494. Springer, Heidelberg (2005).  https://doi.org/10.1007/11593447_26CrossRefGoogle Scholar
  29. 29.
    Maurer, U.M., Tessaro, S.: A hardcore lemma for computational indistinguishability: security amplification for arbitrarily weak PRGs with optimal stretch. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 237–254. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-11799-2_15CrossRefGoogle Scholar
  30. 30.
    Mendel, F., Rechberger, C., Schläffer, M.: MD5 is weaker than weak: attacks on concatenated combiners. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 144–161. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-10366-7_9CrossRefGoogle Scholar
  31. 31.
    Mittelbach, A.: Cryptophia’s short combiner for collision-resistant hash functions. In: Jacobson, M., Locasto, M., Mohassel, P., Safavi-Naini, R. (eds.) ACNS 2013. LNCS, vol. 7954, pp. 136–153. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38980-1_9CrossRefGoogle Scholar
  32. 32.
    Moshkovitz, D., Barak, B.: Communication complexity. Scribes for Advanced Complexity Theory (2012). https://people.csail.mit.edu/dmoshkov/courses/adv-comp/
  33. 33.
    Numayama, A., Isshiki, T., Tanaka, K.: Security of digital signature schemes in weakened random Oracle models. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 268–287. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78440-1_16CrossRefGoogle Scholar
  34. 34.
    Reingold, O., Trevisan, L., Vadhan, S.P.: Notions of reducibility between cryptographic primitives. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 1–20. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24638-1_1CrossRefzbMATHGoogle Scholar
  35. 35.
    Rogaway, P., Shrimpton, T.: Cryptographic hash-function basics: definitions, implications, and separations for preimage resistance, second-preimage resistance, and collision resistance. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 371–388. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-25937-4_24CrossRefGoogle Scholar
  36. 36.
    Simon, D.R.: Finding collisions on a one-way street: can secure hash functions be based on general assumptions? In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 334–345. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0054137CrossRefGoogle Scholar
  37. 37.
    Unruh, D.: Random oracles and auxiliary input. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 205–223. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74143-5_12CrossRefGoogle Scholar
  38. 38.
    Yao, A.C.-C.: Some complexity questions related to distributive computing (preliminary report). In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, pp. 209–213 (1979)Google Scholar

Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  • Balthazar Bauer
    • 1
    • 2
  • Pooya Farshim
    • 1
    • 2
  • Sogol Mazaheri
    • 3
  1. 1.DI/ENS, CNRS, PSL UniversityParisFrance
  2. 2.InriaParisFrance
  3. 3.CryptoplexityTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations