Following Forrelation – quantum algorithms in exploring Boolean functions' spectra
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Published:2022
Issue:0
Volume:0
Page:0
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ISSN:1930-5346
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Container-title:Advances in Mathematics of Communications
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language:
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Short-container-title:AMC
Author:
Dutta Suman,Maitra Subhamoy,Mukherjee Chandra Sekhar
Abstract
<p style='text-indent:20px;'>Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation spectrum, and the autocorrelation spectrum. We introduce the existing 2-fold Forrelation formulation with bent duality-based promise problems as desirable instantiations. Next, we concentrate on the 3-fold version through two approaches. First, we judiciously set up some of the functions in 3-fold Forrelation so that given oracle access, one can sample from the Walsh Spectrum of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Using this, we obtain improved results than what one can achieve by exploiting the Deutsch-Jozsa algorithm. In turn, it has implications in resiliency checking. Furthermore, we use a similar idea to obtain a technique in estimating the cross-correlation (and thus autocorrelation) value at any point, improving upon the existing algorithms. Finally, we tweak the quantum algorithm with the superposition of linear functions to obtain a cross-correlation sampling technique. This is the first cross-correlation sampling algorithm with constant query complexity to the best of our knowledge. This also provides a strategy to check if two functions are uncorrelated of degree <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula>. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>.</p>
Publisher
American Institute of Mathematical Sciences (AIMS)
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Computer Networks and Communications,Algebra and Number Theory,General Earth and Planetary Sciences,General Engineering,General Environmental Science
Reference21 articles.
1. S. Aaronson, BQP and the polynomial hierarchy, In Proceedings of the Forty-Second ACM Symposium on Theory of Computing (STOC '10), Association for Computing Machinery, New York, NY, USA, (2010), 141–150, arXiv version, arXiv: 0910.4698, (2009). 2. S. Aaronson and A. Ambainis, Forrelation: A problem that optimally separates quantum from classical computing, Siam J. Comput., 47 (2018), 982–1038. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC '15), Association for Computing Machinery, New York, NY, USA, DOI: https://doi.org/10.1145/2746539.2746547, (2015), 307–316, arXiv version, arXiv: 1411.5729, (2014). 3. N. Bansal and M. Sinha, k-forrelation optimally separates quantum and classical query complexity, In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC '21), (2021), 1303–1316. arXiv version, arXiv: 2008.07003, (2020). 4. D. Bera, S. Maitra and S. Tharrmashastha, Efficient quantum algorithms related to autocorrelation spectrum, In Progress in Cryptology-Indocrypt 2019, Lecture Notes in Computer Science, vol. 11898,415–432, Springer, (2019), arXiv version, arXiv: 1808.04448, (2019). 5. G. Brassard, P. Hoyer, M. Mosca and A. Tapp, Quantum amplitude amplification and estimation, Quantum Computation and Quantum Information, Samuel J. Lomonaco, Jr. (editor), AMS Contemporary Mathematics, vol. 305 (2002), 53–74, arXiv version, arXiv: quant-ph/0005055, (2000).
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2 articles.
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