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Following Forrelation -- Quantum Algorithms in Exploring Boolean Functions' Spectra

Suman Dutta, Subhamoy Maitra, Chandra Sekhar Mukherjee

TL;DR

The paper develops a unified quantum Forrelation framework to study spectra of Boolean functions, focusing on Walsh, cross-correlation, and autocorrelation spectra. By using 2-fold Forrelation with bent duality and a 3-fold Forrelation paradigm (with f1=f3=f and a designed f2), it achieves constant-query improvements for Walsh-spectrum sampling and resiliency checks, and introduces cross-correlation sampling algorithms that outperform prior methods. It also leverages superpositions of linear functions and Dicke states to realize efficient cross-correlation estimation and uncorrelated-degree-m testing, framing Forrelation as a versatile tool for cryptographic spectral analysis. The results extend the applicability of Forrelation beyond separation results, providing practical quantum algorithms for spectral estimation and sampling with potential quantum advantage in cryptographic applications.

Abstract

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 an oracle access, one can sample from the Walsh Spectrum of $f$. Using this, we obtain improved results than what we obtain from the Deutsch-Jozsa algorithm, and in turn it has implications 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 $m$.

Following Forrelation -- Quantum Algorithms in Exploring Boolean Functions' Spectra

TL;DR

The paper develops a unified quantum Forrelation framework to study spectra of Boolean functions, focusing on Walsh, cross-correlation, and autocorrelation spectra. By using 2-fold Forrelation with bent duality and a 3-fold Forrelation paradigm (with f1=f3=f and a designed f2), it achieves constant-query improvements for Walsh-spectrum sampling and resiliency checks, and introduces cross-correlation sampling algorithms that outperform prior methods. It also leverages superpositions of linear functions and Dicke states to realize efficient cross-correlation estimation and uncorrelated-degree-m testing, framing Forrelation as a versatile tool for cryptographic spectral analysis. The results extend the applicability of Forrelation beyond separation results, providing practical quantum algorithms for spectral estimation and sampling with potential quantum advantage in cryptographic applications.

Abstract

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 -fold version through two approaches. First, we judiciously set-up some of the functions in -fold Forrelation, so that given an oracle access, one can sample from the Walsh Spectrum of . Using this, we obtain improved results than what we obtain from the Deutsch-Jozsa algorithm, and in turn it has implications 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 . We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of .

Paper Structure

This paper contains 15 sections, 21 theorems, 27 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization and Contribution
  3. Preliminaries and Background
  4. Some Properties of Boolean function
  5. Quantum algorithms of interest
  6. Warming up: $2$-fold Forrelation and bent Duality
  7. Algorithms related to Walsh Spectrum via $3$-Fold Forrelation
  8. A suitable representation of $3$-fold Forrelation and the corresponding algorithms
  9. Walsh Spectrum Sampling using $3$-fold Forrelation
  10. Implications to Resiliency checking
  11. Cross-correlation and $3$-fold Forrelation
  12. Comparison of auto-correlation results with bera
  13. A cross-Correlation sampling algorithm
  14. Checking if $f$ and $g$ are Uncorrelated of degree $m$
  15. Conclusion

Key Result

Theorem 1

The problem of determining $\lvert*\rvert{\Phi_{f,g}} \leq \frac{1}{100}$ or $\Phi_{f,g} \geq \frac{3}{5}$ (under the promise that it is one of the two cases) can be solved in the bounded error quantum query model with a single $query$ whereas any randomized classical algorithm would require at leas

Figures (3)

  • Figure 1: Quantum circuit for implementing the $2$-fold Forrelation problem using $2$ queries.
  • Figure 4: Sampling probabilities of Walsh transform using different algorithms
  • Figure 5: Quantum circuit for implementing Algorithm \ref{['algo']}.

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6: forr
  • Theorem 1: aaron
  • Theorem 2: bera
  • Lemma 1: bera
  • Lemma 2: brass
  • ...and 31 more