Deterministic simulation in LOGSPACE
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
One-way functions and Pseudorandom generators
Combinatorica - Theory of Computing
Computational Complexity
Randomized Distributed Edge Coloring via an Extension of the Chernoff--Hoeffding Bounds
SIAM Journal on Computing
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Direct product results and the GCD problem, in old and new communication models
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
A Chernoff Bound for Random Walks on Expander Graphs
SIAM Journal on Computing
SIAM Journal on Computing
Median bounds and their application
Journal of Algorithms
Hard-core distributions for somewhat hard problems
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Does Parallel Repetition Lower the Error in Computationally Sound Protocols?
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Towards proving strong direct product theorems
Computational Complexity
Key agreement from weak bit agreement
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Parallel repetition: simplifications and the no-signaling case
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Perfect Parallel Repetition Theorem for Quantum XOR Proof Systems
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Parallel repetition in projection games and a concentration bound
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Theory and application of trapdoor functions
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
Randomness-Efficient Sampling within NC1
Computational Complexity
Products and help bits in decision trees
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
A Counterexample to Strong Parallel Repetition
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Chernoff-Type Direct Product Theorems
Journal of Cryptology
Approximate List-Decoding of Direct Product Codes and Uniform Hardness Amplification
SIAM Journal on Computing
A Probabilistic Inequality with Applications to Threshold Direct-Product Theorems
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
A Parallel Repetition Theorem for Any Interactive Argument
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Parallel repetition of computationally sound protocols revisited
TCC'07 Proceedings of the 4th conference on Theory of cryptography
CAPTCHA: using hard AI problems for security
EUROCRYPT'03 Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques
Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized
SIAM Journal on Computing
Parallel repetition theorems for interactive arguments
TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
Almost optimal bounds for direct product threshold theorem
TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
Hardness amplification of weakly verifiable puzzles
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
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We give a combinatorial proof of the Chernoff-Hoeffding concentration bound [9,16], which says that the sum of independent {0, 1}- valued random variables is highly concentrated around the expected value. Unlike the standard proofs, our proof does not use the method of higher moments, but rather uses a simple and intuitive counting argument. In addition, our proof is constructive in the following sense: if the sum of the given random variables is not concentrated around the expectation, then we can efficiently find (with high probability) a subset of the random variables that are statistically dependent. As simple corollaries, we also get the concentration bounds for [0, 1]-valued random variables and Azuma's inequality for martingales [4]. We interpret the Chernoff-Hoeffding bound as a statement about Direct Product Theorems. Informally, a Direct Product Theorem says that the complexity of solving all k instances of a hard problem increases exponentially with k; a Threshold Direct Product Theorem says that it is exponentially hard in k to solve even a significant fraction of the given k instances of a hard problem. We show the equivalence between optimal Direct Product Theorems and optimal Threshold Direct Product Theorems. As an application of this connection, we get the Chernoff bound for expander walks [12] from the (simpler to prove) hitting property [2], as well as an optimal (in a certain range of parameters) Threshold Direct Product Theorem for weakly verifiable puzzles from the optimal Direct Product Theorem [8]. We also get a simple constructive proof of Unger's result [38] saying that XOR Lemmas imply Threshold Direct Product Theorems.