Following Forrelation – Quantum Algorithms in Exploring Boolean Functions' Spectra
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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 first study the 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 an oracle access, one can sample from the Walsh Spectrum of f. Using this, we obtain improved results in resiliency checking. Furthermore, we use 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 superposition of linear functions to obtain a cross-correlation sampling technique. To the best of our knowledge, this is the first cross-correlation sampling algorithm with constant query complexity. This also provides a strategy to check if two functions are uncorrelated of degree m. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of m.
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