Threshold computation and cryptographic security

Han, Yenjo; Hemaspaandra, Lane A.; Thierauf, Thomas

doi:10.1007/3-540-57568-5_253

Cited by 38 publications

(52 citation statements)

References 25 publications

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“…PostBPP is easily seen to equal the complexity class BPP path , which was defined by Han, Hemaspaandra, and Thierauf [31]. In particular, it follows from Han et al's results that MA ⊆ PostBPP and that P NP || ⊆ PostBPP ⊆ BPP NP || , where P NP || and BPP NP || denote P and BPP respectively with nonadaptive queries to an NP oracle.…”

Section: Preliminariesmentioning

confidence: 92%

The computational complexity of linear optics

Aaronson

Архипов

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

884

1,578

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We give new evidence that quantum computers-moreover, rudimentary quantum computers built entirely out of linear-optical elements-cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions.Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P #P = BPP NP , and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the Permanent-of-Gaussians Conjecture, which says that it is #P-hard to approximate the permanent of a matrix A of independent N (0, 1) Gaussian entries, with high probability over A; and the Permanent Anti-Concentration Conjecture, which says that |Per (A)| ≥ √ n!/ poly (n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

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Section: Preliminariesmentioning

confidence: 92%

The computational complexity of linear optics

Aaronson

Архипов

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

884

1,578

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“…In contrast to cc 15 (2007) BPP, BPP path is quite powerful: it is known to contain P NP || (Han et al 1997). …”

Section: 4mentioning

confidence: 97%

“…The class BPP path was defined by Han et al (1997). It is the class of languages L for which there exists a non-deterministic polynomial-time Turing Machine M for which…”

Section: 4mentioning

confidence: 99%

Pseudorandomness for Approximate Counting and Sampling

Shaltiel

Umans

2006

comput. complex.

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Abstract. We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to "boost" a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize ArthurMerlin games (i.e., show AM = NP) are in fact all equivalent. We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the "boosting" theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM. We observe that Cai's proof that S P 2 ⊆ ZPP NP and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice.

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“…As an aside, fix a nice multi-arity pairing function · · · (see, e.g., [HHT97]). This function will of course take as arguments tuples of strings (but in fact, it is legal for, as is the case in the definition of L j for example, some components to be natural numbers, in which case the standard bijection between N and Σ * will be implicitly applied) and will output a single string.…”

Section: Up = Ppmentioning

confidence: 99%

The consequences of eliminating NP solutions

Faliszewski

Hemaspaandra

2008

Computer Science Review

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Given a function based on the computation of an NP machine, can one in general eliminate some solutions? That is, can one in general decrease the ambiguity? This simple question remains, even after extensive study by many researchers over many years, mostly unanswered. However, complexity-theoretic consequences and enabling conditions are known. In this tutorial-style article we look at some of those, focusing on the most natural framings: reducing the number of solutions of NP functions, refining the solutions of NP functions, and subtracting from or otherwise shrinking #P functions.We will see how small advice strings are important here, but we also will see how increasing advice size to achieve robustness is central to the proof of a key ambiguityreduction result for NP functions.

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Threshold computation and cryptographic security

Cited by 38 publications

References 25 publications

The computational complexity of linear optics

The computational complexity of linear optics

Pseudorandomness for Approximate Counting and Sampling

The consequences of eliminating NP solutions

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