Relative to a Random Oracle<i>A</i>, ${\bf P}^A \ne {\bf NP}^A \ne \text{co-}{\bf NP}^A $ with Probability 1

Bennett, Charles H.; Gill, John

doi:10.1137/0210008

Cited by 369 publications

(104 citation statements)

References 15 publications

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“…Bennett and Gill prove in [7] that relative to a random oracle the complexity classes BPP and P are equivalent. Let us quickly sketch their idea.…”

Section: Lemma 2 Let G π Be a Construction With Black-box Access To mentioning

confidence: 99%

“…Using the techniques developed by Bennet and Gill [7] we now show that in the multi-stage indifferentiability setting, if a simulator is pseudo deterministic for a distinguisher D, then it can be derandomized, in case the constructed primitive Π is a random oracle or an ideal cipher. When applied to a simulator S that is universal for all distinguishers (strong indifferentiability), these derandomization techniques yield a family of simulators that depends only on the number of queries made by the distinguisher (weak indifferentiability).…”

Section: Lemma 2 Let G π Be a Construction With Black-box Access To mentioning

confidence: 99%

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Reset Indifferentiability and Its Consequences

Baecher

Brzuska

Mittelbach

2013

Advances in Cryptology - ASIACRYPT 2013

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Abstract. The equivalence of the random-oracle model and the idealcipher model has been studied in a long series of results. Holenstein, Künzler, and Tessaro (STOC, 2011) have recently completed the picture positively, assuming that, roughly speaking, equivalence is indifferentiability from each other. However, under the stronger notion of reset indifferentiability this picture changes significantly, as Demay et al. (EU-ROCRYPT, 2013) and Luykx et al. (ePrint, 2012) demonstrate.We complement these latter works in several ways. First, we show that any simulator satisfying the reset indifferentiability notion must be stateless and pseudo deterministic. Using this characterization we show that, with respect to reset indifferentiability, two ideal models are either equivalent or incomparable, that is, a model cannot be strictly stronger than the other model. In the case of the random-oracle model and the ideal-cipher model, this implies that the two are incomparable. Finally, we examine weaker notions of reset indifferentiability that, while not being able to allow composition in general, allow composition for a large class of multi-stage games. Here we show that the seemingly much weaker notion of 1-reset indifferentiability proposed by Luykx et al. is equivalent to reset indifferentiability. Hence, the impossibility of coming up with a reset-indifferentiable construction transfers to the setting where only one reset is permitted, thereby re-opening the quest for an achievable and meaningful notion in between the two variants.

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“…Bennett and Gill prove in [7] that relative to a random oracle the complexity classes BPP and P are equivalent. Let us quickly sketch their idea.…”

Section: Lemma 2 Let G π Be a Construction With Black-box Access To mentioning

confidence: 99%

Section: Lemma 2 Let G π Be a Construction With Black-box Access To mentioning

confidence: 99%

Reset Indifferentiability and Its Consequences

Baecher

Brzuska

Mittelbach

2013

Advances in Cryptology - ASIACRYPT 2013

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“…, x n ) = 0, 10 The reader is encouraged to find such an algorithm. 11 Berger used a set of 20 426 tiles; R. Robinson was able to reduce this number to six, and R. Penrose to two. See more in [74].…”

Section: The World Of Polynomialsmentioning

confidence: 99%

“…To complete the picture we quote the following two results: 2) [11] If A is a random oracle, then P (A) = NP (A), i.e. with probability one P (A) = NP (A).…”

Section: More About P =?Npmentioning

confidence: 99%

Complexity: A Language-Theoretic Point of View

Calude¹,

Hromkovič²

1997

Handbook of Formal Languages

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“…In a multi-stage setting the various instances of the simulator must, however, answer queries consistently, that is, in particular the same query by different adversaries must always be answered with the same answer independent of the order of queries. For this, we build on a derandomization technique developed by Bennet and Gill to show that the complexity classes BPP and P are identical relative to a random oracle [BG81]. One interesting intermediary result is that of a generic indifferentiability simulator that answers queries in a very restricted way.…”

Section: Introductionmentioning

confidence: 99%

Salvaging Indifferentiability in a Multi-stage Setting

Mittelbach

2014

Advances in Cryptology – EUROCRYPT 2014

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Abstract. The indifferentiability framework by Maurer, Renner and Holenstein (MRH; TCC 2004) formalizes a sufficient condition to safely replace a random oracle by a construction based on a (hopefully) weaker assumption such as an ideal cipher. Indeed, many indifferentiable hash functions have been constructed and could since be used in place of random oracles. Unfortunately, Ristenpart, Shacham, and Shrimpton (RSS; Eurocrypt 2011) discovered that for a large class of security notions, the MRH composition theorem actually does not apply. To bridge the gap they suggested a stronger notion called reset indifferentiability and established a generalized version of the MRH composition theorem. However, as recent works by Demay et al. (Eurocrypt 2013) and Baecher et al. (Asiacrypt 2013) brought to light, reset indifferentiability is not achievable thereby re-opening the quest for a notion that is sufficient for multi-stage games and achievable at the same time.We present a condition on multi-stage games that we call unsplittability. We show that if a game is unsplittable for a hash construction then the MRH composition theorem can be salvaged.

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Relative to a Random OracleA, ${\bf P}^A \ne {\bf NP}^A \ne \text{co-}{\bf NP}^A $ with Probability 1

Cited by 369 publications

References 15 publications

Reset Indifferentiability and Its Consequences

Reset Indifferentiability and Its Consequences

Complexity: A Language-Theoretic Point of View

Salvaging Indifferentiability in a Multi-stage Setting

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