Almost Optimal Bounds for Direct Product Threshold Theorem

  • Charanjit S. Jutla
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5978)


We consider weakly-verifiable puzzles which are challenge-response puzzles such that the responder may not be able to verify for itself whether it answered the challenge correctly. We consider k-wise direct product of such puzzles, where now the responder has to solve k puzzles chosen independently in parallel. Canetti et al have earlier shown that such direct product puzzles have a hardness which rises exponentially with k. In the threshold case addressed in Impagliazzo et al, the responder is required to answer correctly a fraction of challenges above a threshold. The bound on hardness of this threshold parallel version was shown to be similar to Chernoff bound, but the constants in the exponent are rather weak. Namely, Impagliazzo et al show that for a puzzle for which probability of failure is δ, the probability of failing on less than (1 − γ)δk out of k puzzles, for any parallel strategy, is at most \(e^{-\gamma^2\delta k/40}\).

In this paper, we develop new techniques to bound this probability, and show that it is arbitrarily close to Chernoff bound. To be precise, the bound is \(e^{-\gamma^2(1-\gamma) \delta k/2}\). We show that given any responder that solves k parallel puzzles with a good threshold, there is a uniformized parallel solver who has the same threshold of solving k parallel puzzles, while being oblivious to the permutation of the puzzles. This enhances the analysis considerably, and may be of independent interest.


  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method. John Wiley and Sons, Chichester (1992)zbMATHGoogle Scholar
  2. 2.
    Canetti, R., Halevi, S., Steiner, M.: Hardness Amplification of Weakly-Verifiable Puzzles. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 17–33. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Holenstein, T.: Parallel Repetition: Simplifications and the No-Signalling Case. In: Proc. ACM STOC (2007)Google Scholar
  4. 4.
    Impagliazzo, R., Jaiswal, R., Kabanets, V.: Chernoff-Type Direct Product Theorem. J. Cryptology 22, 75–93 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Impagliazzo, R., Jaiswal, R., Kabanets, V., Wigderson, A.: Uniform direct-product theorems: Simplified, optimized, and de-randomized. In: Proc. ACM STOC (2008)Google Scholar
  6. 6.
    Knuth, D.: Art of Computer Programming, vol. 1. Addison Wesley, Reading (1973)zbMATHGoogle Scholar
  7. 7.
    Raz, R.: A Parallel Repetition Theorem. SIAM J. of Computing 27(3), 763–803Google Scholar
  8. 8.
    von Ahn, L., Blum, M., Hopper, N.J., Langford, J.: CAPTCHA: Using hard AI problems for security. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 294–311. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Charanjit S. Jutla
    • 1
  1. 1.IBM T. J. Watson Research CenterYorktown Heights

Personalised recommendations