Tomoyuki Yamakami

This paper studies distributions which

can be ``approximated'' by sampling algorithms in time polynomial in

the length of their outputs. First, it is known that if

polynomial-time samplable distributions are polynomial-time

computable, then NP collapses to P. This paper shows by a simple

...
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Miklos Ajtai

We give a random class of n dimensional lattices so that, if

there is a probabilistic polynomial time algorithm which finds a short

vector in a random lattice with a probability of at least 1/2

then there is also a probabilistic polynomial time algorithm which

solves the following three ...
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Miklos Ajtai, Cynthia Dwork

We present a probabilistic public key cryptosystem which is

secure unless the following worst-case lattice problem can be solved in

polynomial time:

"Find the shortest nonzero vector in an n dimensional lattice

L where the shortest vector v is unique in the sense that any other

vector whose ...
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Johannes Köbler, Rainer Schuler

We use the assumption that all sets in NP (or other levels

of the polynomial-time hierarchy) have efficient average-case

algorithms to derive collapse consequences for MA, AM, and various

subclasses of P/poly. As a further consequence we show for

C in {P(PP), PSPACE} that ...
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Shin Aida, Rainer Schuler, Tatsuie Tsukiji, Osamu Watanabe

In this paper we separate many-one reducibility from truth-table

reducibility for distributional problems in DistNP under the

hypothesis that P neq NP. As a first example we consider the

3-Satisfiability problem (3SAT) with two different distributions

on 3CNF formulas. We show that 3SAT using a version of the

standard distribution ...
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Oded Goldreich, Avi Wigderson

For every $\epsilon>0$,

we present a {\em deterministic}\/ log-space algorithm

that correctly decides undirected graph connectivity

on all but at most $2^{n^\epsilon}$ of the $n$-vertex graphs.

The same holds for every problem in Symmetric Log-space (i.e., $\SL$).

Making no assumptions (and in particular not assuming the ... more >>>

Daniele Micciancio

Lattices have received considerable attention as a potential source of computational hardness to be used in cryptography, after a breakthrough result of Ajtai (STOC 1996) connecting the average-case and worst-case complexity of various lattice problems. The purpose of this paper is twofold. On the expository side, we present a rigorous ... more >>>

Luca Trevisan

Error-correcting codes and related combinatorial constructs

play an important role in several recent (and old) results

in computational complexity theory. In this paper we survey

results on locally-testable and locally-decodable error-correcting

codes, and their applications to complexity theory and to

cryptography.

Locally decodable codes are error-correcting codes ... more >>>

Alexander Healy, Salil Vadhan, Emanuele Viola

We revisit the problem of hardness amplification in $\NP$, as

recently studied by O'Donnell (STOC `02). We prove that if $\NP$

has a balanced function $f$ such that any circuit of size $s(n)$

fails to compute $f$ on a $1/\poly(n)$ fraction of inputs, then

$\NP$ has a function $f'$ such ...
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Andrej Bogdanov, Luca Trevisan

We show that if an NP-complete problem has a non-adaptive

self-corrector with respect to a samplable distribution then

coNP is contained in NP/poly and the polynomial

hierarchy collapses to the third level. Feigenbaum and

Fortnow (SICOMP 22:994-1005, 1993) show the same conclusion

under the stronger assumption that an

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Lance Fortnow, Luis Antunes

We show that under a reasonable hardness assumptions, the time-bounded Kolmogorov distribution is a universal samplable distribution. Under the same assumption we exactly characterize the worst-case running time of languages that are in average polynomial-time over all P-samplable distributions.

more >>>Andrej Bogdanov, Luca Trevisan

We survey the theory of average-case complexity, with a

focus on problems in NP.

Parikshit Gopalan, Venkatesan Guruswami

We study the average-case hardness of the class NP against

deterministic polynomial time algorithms. We prove that there exists

some constant $\mu > 0$ such that if there is some language in NP

for which no deterministic polynomial time algorithm can decide L

correctly on a $1- (log n)^{-\mu}$ fraction ...
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Andrej Bogdanov, Muli Safra

An errorless heuristic is an algorithm that on all inputs returns either the correct answer or the special symbol "I don't know." A central question in average-case complexity is whether every distributional decision problem in NP has an errorless heuristic scheme: This is an algorithm that, for every δ > ... more >>>

Edward Hirsch, Dmitry Itsykson

We assume the existence of a function f that is computable in polynomial time but its inverse function is not computable in randomized average-case polynomial time. The cryptographic setting is, however, different: even for a weak one-way function, every possible adversary should fail on a polynomial fraction of inputs. Nevertheless, ... more >>>

Jeffrey C. Jackson, Homin Lee, Rocco Servedio, Andrew Wan

We give an algorithm that with high probability properly learns random monotone t(n)-term

DNF under the uniform distribution on the Boolean cube {0, 1}^n. For any polynomially bounded function t(n) <= poly(n) the algorithm runs in time poly(n, 1/eps) and with high probability outputs an eps accurate monotone DNF ...
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Dmitry Itsykson

We study class AvgBPP that consists of distributional problems that can be solved in average polynomial time (in terms of Levin's average-case complexity) by randomized algorithms with bounded error. We prove that there exists a distributional problem that is complete for AvgBPP under polynomial-time samplable distributions. Since we use deterministic ... more >>>

Akinori Kawachi, Osamu Watanabe

Impagliazzo and Levin demonstrated [IL90] that the average-case hardness of any NP-search problem under any P-samplable distribution implies that of another NP-search problem under the uniform distribution. For this they developed a way to define a reduction from an NP-search problem F with ``mild hardness'' under any P-samplable distribution H; ... more >>>