We investigate the Goldreich-Levin Theorem in the context of quantum information. This result is a reduction from the computational 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-between quantum one-way functions and quantum hard-predicates-is quantitatively more efficient than the known classical version. Roughly speaking, if the one-way function acts on n-bit strings then the overhead in the reduction is by a factor of O(n/ε 2 ) 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. Moreover, we prove via a lower bound that, in a black-box framework, the classical version of the reduction cannot have overhead less than Ω(n/ε 2 ). We also show that, using this reduction, a quantum bit commitment scheme that is perfectly binding and computationally concealing can be obtained from any quantum one-way permutation. This complements a recent result by Dumais, Mayers and Salvail, where the bit commitment scheme is perfectly concealing and computationally binding. We also show how to perform qubit commitment by a similar approach.
The result of REF shows that this reduction can be obtained more efficiently when considering quantum functions and quantum hard-predicates.
A quantum Goldreich-Levin theorem with cryptographic applications
In Proc. of STACS ’02, LNCS 2285
Mathematics Physics Computer Science