SIAM Journal on Computing
Parallel repetition: simplifications and the no-signaling case
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Uniform direct product theorems: simplified, optimized, and derandomized
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Chernoff-Type Direct Product Theorems
Journal of Cryptology
CAPTCHA: using hard AI problems for security
EUROCRYPT'03 Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques
Hardness amplification of weakly verifiable puzzles
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
Constructive proofs of concentration bounds
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Distinguishing distributions using Chernoff information
ProvSec'10 Proceedings of the 4th international conference on Provable security
General hardness amplification of predicates and puzzles
TCC'11 Proceedings of the 8th conference on Theory of cryptography
Counterexamples to hardness amplification beyond negligible
TCC'12 Proceedings of the 9th international conference on Theory of Cryptography
A Parallel Repetition Theorem for Constant-Round Arthur-Merlin Proofs
ACM Transactions on Computation Theory (TOCT)
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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.