Compression from Collisions, or Why CRHF Combiners Have a Long Output

  • Krzysztof Pietrzak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5157)

Abstract

A black-box combiner for collision resistant hash functions (CRHF) is a construction which given black-box access to two hash functions is collision resistant if at least one of the components is collision resistant.

In this paper we prove a lower bound on the output length of black-box combiners for CRHFs. The bound we prove is basically tight as it is achieved by a recent construction of Canetti et al [Crypto’07]. The best previously known lower bounds only ruled out a very restricted class of combiners having a very strong security reduction: the reduction was required to output collisions for both underlying candidate hash-functions given a single collision for the combiner (Canetti et al [Crypto’07] building on Boneh and Boyen [Crypto’06] and Pietrzak [Eurocrypt’07]).

Our proof uses a lemma similar to the elegant “reconstruction lemma” of Gennaro and Trevisan [FOCS’00], which states that any function which is not one-way is compressible (and thus uniformly random function must be one-way). In a similar vein we show that a function which is not collision resistant is compressible. We also borrow ideas from recent work by Haitner et al. [FOCS’07], who show that one can prove the reconstruction lemma even relative to some very powerful oracles (in our case this will be an exponential time collision-finding oracle).

References

  1. 1.
    Barak, B.: How to go beyond the black-box simulation barrier. In: 42nd FOCS, pp. 106–115. IEEE Computer Society Press, Los Alamitos (2001)Google Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Gennaro, R., Trevisan, L.: Lower bounds on the efficiency of generic cryptographic constructions. In: 41st FOCS, pp. 305–313. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  6. 6.
    Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity for all languages in NP have zero-knowledge proof systems. Journal of the. ACM 38(3), 691–729 (1991)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Haitner, I., Hoch, J.J., Reingold, O., Segev, G.: Finding collisions in interactive protocols: a tight lower bound on the round complexity of statistically-hiding commitments. In: 48th FOCS, pp. 669–679. IEEE Computer Society Press, Los Alamitos (2007)Google Scholar
  8. 8.
    Harnik, D., Kilian, J., Naor, M., Reingold, O., Rosen, A.: On robust combiners for oblivious transfer and other primitives. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 96–113. Springer, Heidelberg (2005)Google Scholar
  9. 9.
    Herzberg, A.: On tolerant cryptographic constructions. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 172–190. Springer, Heidelberg (2005)Google Scholar
  10. 10.
    Impagliazzo, R., Rudich, S.: Limits on the Provable Consequences of One-way Permutations. In: Proc.  21st ACM STOC, pp. 44–61. ACM Press, New York (1989)Google Scholar
  11. 11.
    Kim, J.H., Simon, D.R., Tetali, P.: Limits on the efficiency of one-way permutation-based hash functions. In: 40th FOCS, pp. 535–542. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  12. 12.
    Lehmann, A., Fischlin, M., Pietrzak, K.: Robust multi-property combiners for hash functions revisited. In: ICALP 2008. LNCS. Springer, Heidelberg (2008)Google Scholar
  13. 13.
    Meier, R., Przydatek, B.: On robust combiners for private information retrieval and other primitives. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 555–569. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Meier, R., Przydatek, B., Wullschleger, J.: Robuster combiners for oblivious transfer. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 404–418. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
  16. 16.
    Pietrzak, K.: Non-trivial black-box combiners for collision-resistant hash-functions don’t exist. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 23–33. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    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)Google Scholar
  18. 18.
    Rogaway, P.: Formalizing human ignorance. In: Nguyên, P.Q. (ed.) VIETCRYPT 2006. LNCS, vol. 4341, pp. 211–228. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    Wang, X., Yin, Y.L., Yu, H.: Finding collisions in the full SHA-1. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 17–36. Springer, Heidelberg (2005)Google Scholar
  21. 21.
    Wang, X., Yu, H.: How to break MD5 and other hash functions. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 19–35. Springer, Heidelberg (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Krzysztof Pietrzak
    • 1
  1. 1.CWI AmsterdamThe Netherlands

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