FOLLOWING FORRELATION – QUANTUM ALGORITHMS IN EXPLORING BOOLEAN FUNCTIONS’ SPECTRA

Article Type

Research Article

Publication Title

Advances in Mathematics of Communications

Abstract

Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptograph-ically 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 f. 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 de-gree m. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of m.

First Page

1

Last Page

25

DOI

10.3934/amc.2021067

Publication Date

2-1-2024

Comments

Open Access; Gold Open Access; Green Open Access

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