We consider the problem to determine the maximal number of satisfiable 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 algorithms and in cryptography. Namely we show that our conjectures imply the following facts:

Our conjectures are strong in that they assume cryptographic hardness: no polynomial algorithm can solve the problem on any non-negligible fraction of inputs. Nevertheless, to the best of our knowledge no efficient algorithms are currently known that refute any of our hardness conjectures.