Pseudo-random permutation generators and cryptographic composition
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
On the construction of pseudo-random permutations: Luby-Rackoff revisited (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Resettable zero-knowledge (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Foundations of Cryptography: Basic Tools
Foundations of Cryptography: Basic Tools
Does Parallel Repetition Lower the Error in Computationally Sound Protocols?
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Resettably-Sound Zero-Knowledge and its Applications
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
An efficient parallel repetition theorem for Arthur-Merlin games
Proceedings of the thirty-ninth 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
Security Amplification for Interactive Cryptographic Primitives
TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
Leakage-Resilient Public-Key Cryptography in the Bounded-Retrieval Model
CRYPTO '09 Proceedings of the 29th Annual International Cryptology Conference on Advances in Cryptology
Computational Indistinguishability Amplification: Tight Product Theorems for System Composition
CRYPTO '09 Proceedings of the 29th Annual International Cryptology Conference on Advances in Cryptology
A Parallel Repetition Theorem for Any Interactive Argument
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Resolving the Simultaneous Resettability Conjecture and a New Non-Black-Box Simulation Strategy
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
Random oracles and auxiliary input
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Amplifying collision resistance: a complexity-theoretic treatment
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Chernoff-type direct product theorems
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Degradation and amplification of computational hardness
TCC'08 Proceedings of the 5th conference on Theory of cryptography
On the Insecurity of Parallel Repetition for Leakage Resilience
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
TCC'11 Proceedings of the 8th conference on Theory of cryptography
Parallel repetition for leakage resilience amplification revisited
TCC'11 Proceedings of the 8th conference on Theory of cryptography
Stateless Cryptographic Protocols
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
An efficient parallel repetition theorem
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
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
Public-Key encryption in the bounded-retrieval model
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
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If we have a problem that is mildly hard, can we create a problem that is significantly harder? A natural approach to hardness amplification is the "direct product"; instead of asking an attacker to solve a single instance of a problem, we ask the attacker to solve several independently generated ones. Interestingly, proving that the direct product amplifies hardness is often highly non-trivial, and in some cases may be false. For example, it is known that the direct product (i.e. "parallel repetition") of general interactive games may not amplify hardness at all. On the other hand, positive results show that the direct product does amplify hardness for many basic primitives such as one-way functions, weakly-verifiable puzzles, and signatures. Even when positive direct product theorems are shown to hold for some primitive, the parameters are surprisingly weaker than what we may have expected. For example, if we start with a weak one-way function that no poly-time attacker can break with probability 1/2, then the direct product provably amplifies hardness to some negligible probability. Naturally, we would expect that we can amplify hardness exponentially, all the way to 2−n probability, or at least to some fixed/known negligible such as n−logn in the security parameter n, just by taking sufficiently many instances of the weak primitive. Although it is known that such parameters cannot be proven via black-box reductions, they may seem like reasonable conjectures, and, to the best of our knowledge, are widely believed to hold. In fact, a conjecture along these lines was introduced in a survey of Goldreich, Nisan and Wigderson (ECCC '95). In this work, we show that such conjectures are false by providing simple but surprising counterexamples. In particular, we construct weakly secure signatures and one-way functions, for which standard hardness amplification results are known to hold, but for which hardness does not amplify beyond just negligible. That is, for any negligible function ε(n), we instantiate these primitives so that the direct product can always be broken with probability ε(n), no matter how many copies we take.