The classical and quantum complexity of the Goldreich-Levin problem with applications to bit commitment

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2004
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Abstract
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.
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Bibliography: p. 125-127
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Citation
Adcock, M. R. (2004). The classical and quantum complexity of the Goldreich-Levin problem with applications to bit commitment (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/12092
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