How to Fake Auxiliary Input

Jetchev, Dimitar; Pietrzak, Krzysztof

doi:10.1007/978-3-642-54242-8_24

Cited by 28 publications

(41 citation statements)

References 25 publications

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“…Also, the proof of Gentry and Wichs goes through the Minimax Theorem (as in Impagliazzo [1995] and Reingold et al [2008]); it would be interesting to give a boosting-style proof (as in Impagliazzo [1995], Klivans and Servedio [2003], Barak et al [2009], Trevisan et al [2009], andZhang [2011]). Jetchev and Pietrzak [2014] proved a certain strengthening of the Dense Model Theorem. What kind of lower bounds can be proven on the complexity of black-box reductions for their result?…”

Section: Open Problemsmentioning

confidence: 95%

“…In addition to the original application of showing that the primes contain arbitrarily long arithmetic progressions, the Dense Model Theorem 1:2 T. Watson has found applications in differential privacy [Mironov et al 2009], pseudoentropy and leakage-resilient cryptography [Barak et al 2003;Reingold et al 2008;Dziembowski and Pietrzak 2008], and graph decompositions [Reingold et al 2008], as well as further applications in additive combinatorics [Gowers and Wolf 2011;2012;Zhao 2013]. Subsequent variants of the Dense Model Theorem have found applications in cryptography [Gentry and Wichs 2011;Jetchev and Pietrzak 2014] and pseudorandomness [Trevisan et al 2009]. …”

Section: Introductionmentioning

confidence: 98%

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Advice Lower Bounds for the Dense Model Theorem

Watson

2015

ACM Trans. Comput. Theory

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We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution D that is δ-dense in a distribution that is -indistinguishable from uniform, there exists a "dense model" for D, that is, a distribution that is δ-dense in the uniform distribution and is -indistinguishable from D. This -indistinguishability is with respect to an arbitrary small class of functions F. For the natural case where ≥ ( δ) and ≥ δ O(1) , our lower bound implies that(1/ ) log(1/δ)·log |F| advice bits are necessary for a certain type of reduction that establishes a stronger form of the Dense Model Theorem (and which encompasses all known proofs of the Dense Model Theorem in the literature). There is only a polynomial gap between our lower bound and the best upper bound for this case (due to Zhang), which is O (1/ 2 ) log(1/δ) · log |F| . Our lower bound can be viewed as an analogue of list size lower bounds for list-decoding of error-correcting codes, but for "dense model decoding" instead. ACM Reference Format:Watson, T. 2014. Advice lower bounds for the dense model theorem.

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Section: Open Problemsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 98%

Advice Lower Bounds for the Dense Model Theorem

Watson

2015

ACM Trans. Comput. Theory

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“…The work of Jetchev and Pietrzak [37] provides a constructive way to simulate the value of the condition, which enables the proof of the chain rule for a relaxed definition of HILL entropy. The work of Vadhan and Zheng [60] provides a proof of the conditional entropy loss result via a uniform reduction, making the result constructive in a very strong sense.…”

Section: Work On Conditional Computational Entropymentioning

confidence: 99%

A Unified Approach to Deterministic Encryption: New Constructions and a Connection to Computational Entropy

2013

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This paper addresses deterministic public-key encryption schemes (DE), which are designed to provide meaningful security when only source of randomness in the encryption process comes from the message itself. We propose a general construction of DE that unifies prior work and gives novel schemes. Specifically, its instantiations include:• The first construction from any trapdoor function that has sufficiently many hardcore bits.• The first construction that provides "bounded" multi-message security (assuming lossy trapdoor functions).The security proofs for these schemes are enabled by three tools that are of broader interest:• A weaker and more precise sufficient condition for semantic security on a high-entropy message distribution. Namely, we show that to establish semantic security on a distribution M of messages, it suffices to establish indistinguishability for all conditional distribution M |E, where E is an event of probability at least 1/4. (Prior work required indistinguishability on all distributions of a given entropy.)• A result about computational entropy of conditional distributions. Namely, we show that conditioning on an event E of probability p reduces the quality of computational entropy by a factor of p and its quantity by log 2 1/p.• A generalization of leftover hash lemma to correlated distributions.We also extend our result about computational entropy to the average case, which is useful in reasoning about leakage-resilient cryptography: leaking λ bits of information reduces the quality of computational entropy by a factor of 2 λ and its quantity by λ.

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“…The quantitative bounds for ε , s in the statement of this lemma are from [JP14]. The result from [VZ13] can be used to get a better s ≈ sε 2 /2 bound (and the same ε ≈ ε), whenever s is large enough as a function of 1/ε and 2 , concretely, this bound holds if s = Ω(2 /ε 4 ).…”

Section: The Counterexamplesmentioning

confidence: 99%

A Counterexample to the Chain Rule for Conditional HILL Entropy

Krenn

Pietrzak

Wadia

2013

Theory of Cryptography

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Most entropy notions H(.) like Shannon or min-entropy satisfy a chain rule stating that for random variables X, Z and A we have H(X|Z, A) ≥ H(X|Z) − |A|. That is, by conditioning on A the entropy of X can decrease by at most the bitlength |A| of A. Such chain rules are known to hold for some computational entropy notions like Yao's and unpredictability-entropy. For HILL entropy, the computational analogue of min-entropy, the chain rule is of special interest and has found many applications, including leakage-resilient cryptography, deterministic encryption and memory delegation. These applications rely on restricted special cases of the chain rule. Whether the chain rule for conditional HILL entropy holds in general was an open problem for which we give a strong negative answer: We construct joint distributions (X, Z, A), where A is a distribution over a single bit, such that the HILL entropy H HILL (X|Z) is large but H HILL (X|Z, A) is basically zero.Our counterexample just makes the minimal assumption that NP P/poly. Under the stronger assumption that injective one-way function exist, we can make all the distributions efficiently samplable.Finally, we show that some more sophisticated cryptographic objects like lossy functions can be used to sample a distribution constituting a counterexample to the chain rule making only a single invocation to the underlying object. * This is the full version of [KPW13] that appeared in TCC 2013.

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How to Fake Auxiliary Input

Cited by 28 publications

References 25 publications

Advice Lower Bounds for the Dense Model Theorem

Advice Lower Bounds for the Dense Model Theorem

A Unified Approach to Deterministic Encryption: New Constructions and a Connection to Computational Entropy

A Counterexample to the Chain Rule for Conditional HILL Entropy

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