The classical Goldreich-Levin (G-L) theorem is presented as a block-box query problem, referred to herein as the G-L problem. The query complexity of this problem is bounded in both classical and quantum settings. The well-known upper bound of the classical G-L problem is analyzed in a pedagogical manner. A proof of the lower bound of the classical G-L problem is given using the techniques of classical information theory. This classical analysis is then extended to the realm of quantum computin g by noting the similarity of the noiseless G-L problem to the inner-product query problem solved by the quantum circuit defined by Bernstein and Vazirani (B-V) . An upper bound of the query complexity of the quantum G-L problem is proven by extension of the B-V circuit to incorporate noisy inner-product queries. The lower bound of the query complexity of the quantum G-L problem is proven by adapting the proof of the optimaliy of the quantum search algorithm to include modified inner-product queries. Both the classical and quantum versions of the Goldreich-Levi n theorem have cryptographic applications in the area of bit commitment. A discussion of the impossibility of unconditional quantum bit commitment is followed by the presentation of both classical and quantum bit commitment protocols that are based on the assumption of the existence of classical and quantum one-way permutations. The relative security of the classical and quant m protocols are compared where it is shown that the quantum version is quantitatively more secure.
We present a quadratic Goldreich–Levin algorithm that is nearly optimal in the following ways. Given a bounded function
f:F2n→R and any
ε>0, the algorithm outputs a quadratic polynomial
q:F2n→F2 whose correlation with
(−1)q is within an additive
ε of the maximum achievable correlation with any quadratic phase function. It runs in
Oε(n3) time and makes
Oε(n2logn) queries to
f, matching the information-theoretic lower bound up to a logarithmic factor. The design of our algorithm draws on ideas from recent advances in quantum learning theory and departs from previous approaches based on algorithmic proofs of the inverse theorem for the Gowers uniformity norms.