Average-Case Complexity

Bogdanov, Andrej; Trevisan, Luca

doi:10.1561/0400000004

Cited by 114 publications

(101 citation statements)

References 83 publications

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“…We will follow the paper [1] of Bogdanov and Trevisan. The main goal of the theory is to show that certain NP problems are "hard on average".…”

Section: Polynomial Time Samplable and Computable Distributionsmentioning

confidence: 99%

“…We will not define here the type of reductions used in the definition of completeness, and refer to [2,4,1] for the definition. We will define simplified reductions (Definition 3.1 from [1]) that come back to [5]. These simplified reductions will suffice for the goal of this paper.…”

Section: Definition 1 ([2 1])mentioning

confidence: 99%

“…Now we are going to define polynomial time computable ensembles of distributions. We will follow the exposition of [1]. Let denote the lexicographic ordering between bit strings.…”

Section: Definition 1 ([2 1])mentioning

confidence: 99%

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Encoding Invariance in Average Case Complexity

Vereshchagin

2013

Theory Comput Syst

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When we represent a decision problem, like CIRCUIT-SAT, as a language over the binary alphabet, we usually do not specify how to encode instances by binary strings. This relies on the empirical observation that the truth of a statement of the form "CIRCUIT-SAT belongs to a complexity class C" does not depend on the encoding, provided both the encoding and the class C are "natural". In this sense most of the Complexity theory is "encoding invariant".The notion of a polynomial time computable distribution from Average Case Complexity is one of the exceptions from this rule. It might happen that a distribution over some objects, like circuits, is polynomial time computable in one encoding and is not polynomial time computable in the other encoding. In this paper we suggest an encoding invariant generalization of a notion of a polynomial time computable distribution. The completeness proofs of known distributional problems, like Bounded Halting, are simpler for the new class than for polynomial time computable distributions. This paper has no new technical contributions. All the statements are proved using the known techniques.

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“…We will follow the paper [1] of Bogdanov and Trevisan. The main goal of the theory is to show that certain NP problems are "hard on average".…”

Section: Polynomial Time Samplable and Computable Distributionsmentioning

confidence: 99%

Section: Definition 1 ([2 1])mentioning

confidence: 99%

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Encoding Invariance in Average Case Complexity

Vereshchagin

2013

Theory Comput Syst

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“…In this subsection, we define the key notions of the average-case complexity theory. Essentially, we follow [2]. We restrict ourselves to the two-letter alphabet {0, 1}; the set of all words in this alphabet is denoted by {0, 1} * .…”

Section: Theorem 3 If There Exists a Length-preserving Polynomial-timentioning

confidence: 99%

“…(1) A successful cryptographic adversary may err on a polynomial fraction of inputs [3, Definition 2.2.2], while an average-case polynomial-time algorithm cannot spend exponential time on a polynomial fraction of inputs [2,6]. (2) A (polynomial-time samplable) probability distribution in the cryptographic setting is taken over the inputs of a function, while in the average-case setting, it is usually taken over the outputs (i.e., over the instances of the problem of inverting the function); see, e.g., [4,7].…”

Section: §1 Introductionmentioning

confidence: 99%

An infinitely-often one-way function based on an average-case assumption

Hirsch

Itsykson

2010

St. Petersburg Math. J.

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Abstract. We assume the existence of a function f that is computable in polynomial time but cannot be inverted by a randomized average-case polynomial algorithm. The cryptographic setting is, however, different: even for a weak one-way function, a successful adversary should fail on a polynomial fraction of inputs. Nevertheless, we show how to construct an infinitely-often one-way function based on f .

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A remark on pseudo proof systems and hard instances of the satisfiability problem

Maly

Müller

2018

Mathematical Logic Qtrly

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We link two concepts from the literature, namely hard sequences for the satisfiability problem sat and so‐called pseudo proof systems proposed for study by Krajíček. Pseudo proof systems are elements of a particular nonstandard model constructed by forcing with random variables. We show that the existence of mad pseudo proof systems is equivalent to the existence of a randomized polynomial time procedure with a highly restrictive use of randomness which produces satisfiable formulas whose satisfying assignments are probably hard to find.

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Average-Case Complexity

Cited by 114 publications

References 83 publications

Encoding Invariance in Average Case Complexity

Encoding Invariance in Average Case Complexity

An infinitely-often one-way function based on an average-case assumption

A remark on pseudo proof systems and hard instances of the satisfiability problem

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