 
Summary: More on average case vs approximation complexity
Michael Alekhnovich
April 13, 2003
Abstract
We consider the problem to determine the maximal number of satisable equations
in a linear system chosen at random. We make several plausible conjectures about the
average case hardness of this problem for some natural distributions on the instances,
and relate them to several interesting questions in the theory of approximation algo
rithms and in cryptography. Namely we show that our conjectures imply the following
facts:
Feige's hypothesis about the hardness of refuting a random 3CNF is true, which in
turn implies inapproximability within a constant for several combinatorial prob
lems, for which no NPhardness of approximation is known.
It is hard to approximate the NEAREST CODEWORD within factor n 1 .
It is hard to estimate the rigidity of a matrix. More exactly, it is hard to distin
guish between matrices of low rigidity and random ones.
There exists a secure publickey (probabilistic) cryptosystem, based on the in
tractability of decoding of random binary codes.
Our conjectures are strong in that they assume cryptographic hardness: no polyno
mial algorithm can solve the problem on any nonnegligible fraction of inputs. Never
