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TR06-154 | 13th December 2006 00:00

Uniform Hardness Amplification in NP via Monotone Codes


Authors: Joshua Buresh-Oppenheim, Valentine Kabanets, Rahul Santhanam
Publication: 15th December 2006 13:59
Downloads: 1421


We consider the problem of amplifying uniform average-case hardness
of languages in $\NP$, where hardness is with respect to $\BPP$
algorithms. We introduce the notion of \emph{monotone}
error-correcting codes, and show that hardness amplification for
$\NP$ is essentially equivalent to constructing efficiently
\emph{locally} encodable and \emph{locally} list-decodable monotone
codes. The previous hardness amplification results for
$\NP$~\cite{Tre03,Tre05} focused on giving a direct construction of
some \emph{locally} encodable/decodable monotone codes, running into
the problem of large amounts of nonuniformity used by the decoding
algorithm. In contrast, we propose the \emph{indirect} approach to
constructing locally encodable/decodable monotone codes, combining
the uniform Direct Product Lemma of~\cite{IJK06} and arbitrary,
\emph{not necessarily locally encodable}, monotone codes. The latter
codes have fewer restrictions, and so may be easier to construct.

We study what parameters are achievable by monotone codes in
general, giving negative and positive results. We present two
constructions of monotone codes. Our first code is a uniquely
decodable code based on the Majority function, and has an efficient
decoding algorithm. Our second code is combinatorially
list-decodable, but we do not have an efficient decoding
algorithm. In conjunction with an appropriate Direct Product Lemma, our
first code yields uniform hardness amplification for $\NP$ from
inverse polynomial to constant average-case hardness. Our second
code, even with a brute-force decoding algorithm, yields further
hardness amplification to $1/2-\log^{-\Omega(1)} n$. Together, these
give an alternative proof of Trevisan's result~\cite{Tre03,Tre05}. Getting any non-brute-force decoding algorithm for our second code
would imply improved parameters for the problem of hardness amplification in $\NP$.

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