A quantum goldreich-levin theorem with cryptographic applications

Abstract

We investigate the Goldreich-Levin Theorem in the context of quantum information. This result is a reduction from the problem of inverting a one-way function to the problem of predicting a particular bit associated with that function. We show that the quantum version of the reduction is quantitatively more efficient than the known classical version. If the one-way function acts on n-bit strings then the overhead in the reduction is by a factor of O(n/ε2) in the classical case but only by a factor of O(1/ε) in the quantum case, where 1/2 +ε is the probability of predicting the hard-predicate. We also show that, using the Goldreich-Levin Theorem, a quantum bit (or qubit) commitment scheme that is perfectly binding and computationally concealing can be obtained from any quantum one-way permutation.