On the Hardness of Computing the Permanent of Random Matrices (Extended Abstract)

We study the complexity of computing the permanent on random inputs. We consider matrices drawn randomly from the space of n by n matrices with integer values between 0 and p –1, for any large enough prime p . We show that any polynomial time algorithm which computes the permanent correctly on even an exponentially small fraction of these matrices, implies the collapse of the polynomial-time hierarchy to its second level. We also show that it is hard to get partial information about the value of the permanent modulo p . We show that any balanced polynomial-time 0/1 predicate (e.g., the least significant bit, the parity of all the bits, the quadratic residuosity character) cannot be guessed with probability significantly greater than 1/2 (unless the polynomial hierarchy collapses). This result can be extended to showing simultaneous hardness for linear size groups of bits.