Balancing Output Length and Query Bound in Hardness Preserving Constructions of Pseudorandom Functions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8885)

Abstract

We revisit hardness-preserving constructions of a pseudo-random function (PRF) from any length doubling pseudo-random generator (PRG) when there is a non-trivial upper bound \(q\) on the number of queries that the adversary can make to the PRF. Very recently, Jain, Pietrzak, and Tentes (TCC 2012) gave a hardness-preserving construction of a PRF that makes only \(O(\log q)\) calls to the underlying PRG when \(q = 2^{n^\epsilon }\) and \(\epsilon \ge \frac{1}{2}\). This dramatically improves upon the efficiency of the construction of Goldreich, Goldwasser, and Micali (FOCS 1984). However, they explicitly left open the question of whether such constructions exist when \(\epsilon < \frac{1}{2}\). In this work, we give constructions of PRFs that make only \(O(\log q)\) calls to the underlying PRG when \(q = 2^{n^\epsilon }\), for \(0<\epsilon <1\); our PRF outputs \(O(n^{2\epsilon })\) bits (on every input), as opposed to the construction of Jain et al. that outputs \(n\) bits. That is, our PRF is not length preserving; however it outputs more bits than the PRF of Jain et al. when \(\epsilon >\frac{1}{2}\). We obtain our construction through the use of information-theoretic tools such as almost\(\alpha \)-wise independent hash functions coupled with a novel proof strategy.

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References

  1. 1.
    Alon, N., Goldreich, O., Håstad, J., Peralta, R.: Simple construction of almost k-wise independent random variables. Random Struct. Algorithms 3(3), 289–304 (1992)CrossRefMATHGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Berman, I., Haitner, I., Komargodski, I., Naor, M.: Hardness preserving reductions via cuckoo hashing. IACR Cryptology ePrint Archive 2012: 722 (2012)Google Scholar
  4. 4.
    Berman, I., Haitner, I., Komargodski, I., Naor, M.: Hardness preserving reductions via cuckoo hashing. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 40–59. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo random bits. In: 23rd Annual Symposium on Foundations of Computer Science, Chicago, Illinois, USA, November 3-5, pp. 112–117 (1982)Google Scholar
  6. 6.
    Carter, L., Wegman, M.N.: Universal classes of hash functions (extended abstract). In: Proceedings of the 9th Annual ACM Symposium on Theory of Computing, Boulder, Colorado, USA, May 4-6, pp. 106–112 (1977)Google Scholar
  7. 7.
    Chandran, N., Garg, S.: Hardness preserving constructions of pseudorandom functions, revisited. IACR Cryptology ePrint Archive 2012: 616 (2012)Google Scholar
  8. 8.
    Cho, C., Lee, C.-K., Ostrovsky, R.: Equivalence of uniform key agreement and composition insecurity. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 447–464. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Goldreich, O.: Towards a theory of software protection. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 426–439. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  10. 10.
    Goldreich, O.: Two remarks concerning the goldwasser-micali-rivest signature scheme. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 104–110. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Kurosawa, K., Johansson, T., Stinson, D.R.: Almost \(k\)-wise independent sample spaces and their cryptologic applications. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 409–421. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  15. 15.
    Levin, L.A.: One-way functions and pseudorandom generators. Combinatorica 7(4), 357–363 (1987)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Luby, M.: Pseudorandomness and cryptographic applications. Princeton computer science notes. Princeton University Press (1996)Google Scholar
  17. 17.
    Luby, M., Rackoff, C.: A study of password security. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 392–397. Springer, Heidelberg (1988)Google Scholar
  18. 18.
    Maurer, U.M.: Indistinguishability of random systems. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 110–133. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    Naor, J., Naor, M.: Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput. 22(4), 838–856 (1993)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Naor, M., Reingold, O.: Synthesizers and their application to the parallel construction of psuedo-random functions. In: 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, October 23-25, pp. 170–181 (1995)Google Scholar
  22. 22.
    Pietrzak, K.: Composition does not imply adaptive security. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 55–65. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Pietrzak, K.: Composition implies adaptive security in minicrypt. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 328–338. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Wegman, M.N., Carter, L.: New classes and applications of hash functions. In: 20th Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, October 29-31, pp. 175–182 (1979)Google Scholar
  25. 25.
    Yao, A.C.-C.: Theory and applications of trapdoor functions (extended abstract). In: 23rd Annual Symposium on Foundations of Computer Science, Chicago, Illinois, USA, November 3-5, pp. 80–91 (1982)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Microsoft ResearchBangaloreIndia
  2. 2.University of CaliforniaCaliforniaBerkeley

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