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 $m$.

翻译：我们在这里重新审视获取Forresulation[Aaronson 等人, 2015] 值的量子算法, 以评价一些已知的波林函数的加密重要光谱, 即 Walsh 频谱、 交叉关系频谱和自动关系频谱。 我们首先研究二倍关系配方, 以弯曲的双向双向性承诺问题为理想的即时推移。 我们接下来通过两种方法集中研究三元倍的版本。 首先, 我们明智地设置了三元倍的量子算法, 以获得一个三元倍的反向关系取样技术。 因此, 根据一个神电极的接入, 我们可以从沃尔什· 斯普特朗( $ff$) 获得样本。 我们使用这个方法, 在弹性检查中获得了更好的结果。 此外, 我们使用一个类似的想法来获得一种技术来估计交叉关系( 因此, 自动关系) 值的量子算法值, 在任何点上, 我们用一个不固定的复杂度来进一步修改。