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
counting ... more >>>

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 lattice ... more >>>

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 length ... more >>>

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 ... more >>>

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 ... more >>>

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 >>>

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 >>>

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 >>>

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 ... more >>>

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
more >>>

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.

We survey the theory of average-case complexity, with a
focus on problems in NP.

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 ... more >>>

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 >>>

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 >>>

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 ... more >>>

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 >>>

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 >>>