Extending Forrelation: Quantum Algorithms Related to Generalized Fourier-Correlation

Abstract: In this paper, we study different cryptographically significant spectra of Boolean functions, including the Walsh-Hadamard, cross-correlation, and autocorrelation. The $2^k$-variation by Stanica [IEEE-IT 2016] is considered here with the formulation for any $m \in \mathbb{N}$. Given this, we present the most generalized version of the Deutsch-Jozsa algorithm, which extends the standard and previously extended versions, thereby encompassing them as special cases. Additionally, we generalize the Forrelation formulation by introducing the $m$-Forrelation and propose various quantum algorithms towards its estimation. In this regard, we explore different strategies in sampling these newly defined spectra using the proposed $m$-Forrelation algorithms and present a comparison of their corresponding success probabilities. Finally, we address the problem related to affine transformations of generalized bent functions and discuss quantum algorithms in identifying the shifts between two bent (or negabent) functions with certain modifications, and compare these with existing results.