Skip to main content

Hardness Preserving Reductions via Cuckoo Hashing

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNSC,volume 7785)

Abstract

A common method for increasing the usability and uplifting the security of pseudorandom function families (PRFs) is to “hash” the inputs into a smaller domain before applying the PRF. This approach, known as “Levin’s trick”, is used to achieve “PRF domain extension” (using a short,e.g,fixed, input length PRF to get a variable-length PRF), and more recently to transform non-adaptive PRFs to adaptive ones. Such reductions, however, are vulnerable to a “birthday attack”: after \(\sqrt{|\mathcal{U}}\) queries to the resulting PRF, where \(\mathcal{U}\) being the hash function range, a collision (i.e., two distinct inputs have the same hash value) happens with high probability. As a consequence, the resulting PRF is insecure against an attacker making this number of queries.

In this work we show how to go beyond the birthday attack barrier, by replacing the above simple hashing approach with a variant of cuckoo hashing — a hashing paradigm typically used for resolving hash collisions in a table, by using two hash functions and two tables, and cleverly assigning each element into one of the two tables. We use this approach to obtain: (i) A domain extension method that requires just two calls to the original PRF, can withstand as many queries as the original domain size and has a distinguishing probability that is exponentially small in the non cryptographic work. (ii) A security-preserving reduction from non-adaptive to adaptive PRFs.

Keywords

  • Hash Function
  • Random Function
  • Function Family
  • Pseudorandom Generator
  • Domain Extension

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.

References

  1. Aiello, W., Venkatesan, R.: Foiling Birthday Attacks in Length-Doubling Transformations - Benes: a non-reversible alternative to Feistel. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 307–320. Springer, Heidelberg (1996)

    CrossRef  Google Scholar 

  2. Arbitman, Y., Naor, M., Segev, G.: Backyard cuckoo hashing: Constant worst-case operations with a succinct representation. In: Proceedings of the 51th Annual Symposium on Foundations of Computer Science (FOCS), pp. 787–796 (2010)

    Google Scholar 

  3. Aumüller, M., Dietzfelbinger, M., Woelfel, P.: Explicit and Efficient Hash Families Suffice for Cuckoo Hashing with a Stash. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 108–120. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  4. Bellare, M., Goldwasser, S.: New Paradigms for Digital Signatures and Message Authentication Based on Non-interactive Zero Knowledge Proofs. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 194–211. Springer, Heidelberg (1990)

    Google Scholar 

  5. Bellare, M., Goldreich, O., Krawczyk, H.: Stateless Evaluation of Pseudorandom Functions: Security beyond the Birthday Barrier. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 270–287. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  6. Berman, I., Haitner, I.: From Non-adaptive to Adaptive Pseudorandom Functions. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 357–368. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  7. Blum, M., Evans, W.S., Gemmell, P., Kannan, S., Naor, M.: Checking the correctness of memories. Algorithmica 12(2/3), 225–244 (1994)

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Carter, L.J., Wegman, M.N.: Universal classes of hash functions. Journal of Computer and System Sciences, 143–154 (1979)

    Google Scholar 

  9. Chandran, N., Garg, S.: Hardness preserving constructions of pseudorandom functions, revisited. IACR Cryptology ePrint Archive 2012:616 (2012)

    Google Scholar 

  10. Chor, B., Fiat, A., Naor, M., Pinkas, B.: Tracing traitors. IEEE Transactions on Information Theory 46(3), 893–910 (2000)

    CrossRef  MATH  Google Scholar 

  11. Dietzfelbinger, M., Woelfel, P.: Almost random graphs with simple hash functions. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pp. 629–638 (2003)

    Google Scholar 

  12. Goldreich, O.: Towards a Theory of Software Protection. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 426–439. Springer, Heidelberg (1987)

    CrossRef  Google Scholar 

  13. Goldreich, O., Goldwasser, S., Micali, S.: On the Cryptographic Applications of Random Functions. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 276–288. Springer, Heidelberg (1985)

    CrossRef  Google Scholar 

  14. Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. Journal of the ACM, 792–807 (1986)

    Google Scholar 

  15. Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM Journal on Computing, 1364–1396 (1999)

    Google Scholar 

  16. Jain, A., Pietrzak, K., Tentes, A.: Hardness Preserving Constructions of Pseudorandom Functions. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 369–382. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  17. Jetchev, D., Özen, O., Stam, M.: Understanding Adaptivity: Random Systems Revisited. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 313–330. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  18. Levin, L.A.: One-way functions and pseudorandom generators. Combinatorica 7(4), 357–363 (1987)

    MathSciNet  CrossRef  MATH  Google Scholar 

  19. Luby, M.: Pseudorandomness and cryptographic applications. Princeton computer science notes. Princeton University Press (1996)

    Google Scholar 

  20. Luby, M., Rackoff, C.: How to construct pseudorandom permutations from pseudorandom functions. SIAM Journal on Computing 17(2), 373–386 (1988)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Maurer, U.M.: Indistinguishability of Random Systems. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 110–132. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  22. Maurer, U.M., Pietrzak, K.: Composition of Random Systems: When Two Weak Make One Strong. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 410–427. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  23. Myers, S.: Black-Box Composition Does Not Imply Adaptive Security. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 189–206. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  24. Nandi, M.: A Unified Method for Improving PRF Bounds for a Class of Blockcipher Based MACs. In: Hong, S., Iwata, T. (eds.) FSE 2010. LNCS, vol. 6147, pp. 212–229. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  25. Naor, M., Reingold, O.: Synthesizers and their application to the parallel construction of psuedo-random functions. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS), pp. 170–181 (1995)

    Google Scholar 

  26. Ostrovsky, R.: An Efficient Software Protection Scheme. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 610–611. Springer, Heidelberg (1990)

    Google Scholar 

  27. Pagh, A., Pagh, R.: Uniform hashing in constant time and optimal space. SIAM Journal on Computing 38(1), 85–96 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  28. Pagh, R., Rodler, F.F.: Cuckoo hashing. J. Algorithms 51(2), 122–144 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  29. Patarin, J.: Security of Random Feistel Schemes with 5 or More Rounds. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 106–122. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  30. Patarin, J.: A Proof of Security in O(2n) for the Benes Scheme. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 209–220. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  31. Patarin, J.: Security of balanced and unbalanced feistel schemes with linear non equalities. IACR Cryptology ePrint Archive, 2010:293 (2010)

    Google Scholar 

  32. Pietrzak, K.: Composition Does Not Imply Adaptive Security. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 55–65. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  33. Pietrzak, K.: Composition Implies Adaptive Security in Minicrypt. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 328–338. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  34. Siegel, A.: On universal classes of extremely random constant-time hash functions. SIAM Journal on Computing 33(3), 505–543 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  35. Wegman, M.N., Carter, L.: New hash functions and their use in authentication and set equality. J. Comput. Syst. Sci. 22(3), 265–279 (1981)

    MathSciNet  CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 International Association for Cryptologic Research

About this paper

Cite this paper

Berman, I., Haitner, I., Komargodski, I., Naor, M. (2013). Hardness Preserving Reductions via Cuckoo Hashing. In: Sahai, A. (eds) Theory of Cryptography. TCC 2013. Lecture Notes in Computer Science, vol 7785. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36594-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-36594-2_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36593-5

  • Online ISBN: 978-3-642-36594-2

  • eBook Packages: Computer ScienceComputer Science (R0)