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Permutation Superposition Oracles for Quantum Query Lower Bounds

Christian Majenz, Giulio Malavolta, Michael Walter

TL;DR

This work extends quantum lower-bound techniques to the random permutation model by introducing a twirled, strictly monotone-factorized permutation oracle (TSPO) and a corresponding Fundamental Lemma. It defines a robust progress-measure framework to bound the probability that a quantum algorithm finds an input-output pair satisfying a relation, culminating in a general bound $\Pr\bigl[(x,\pi(x))\in R\bigr] \leq O\left(\frac{q^3 r_{\max} \ln(N)}{N}\right)$ for $q$ queries. The authors apply this to the one-round sponge, proving unconditional preimage resistance in the random permutation model, and to Unruh’s double-sided zero-search conjecture, providing explicit polynomial-query upper bounds. The results offer a versatile tool for quantum query complexity in permutation settings and set the stage for further generalizations in cryptographic security analyses under quantum models.

Abstract

We propose a generalization of Zhandry's compressed oracle method to random permutations, where an algorithm can query both the permutation and its inverse. We show how to use the resulting oracle simulation to bound the success probability of an algorithm for any predicate on input-output pairs, a key feature of Zhandry's technique that had hitherto resisted attempts at generalization to random permutations. One key technical ingredient is to use strictly monotone factorizations to represent the permutation in the oracle's database. As an application of our framework, we show that the one-round sponge construction is unconditionally preimage resistant in the random permutation model. This proves a conjecture by Unruh.

Permutation Superposition Oracles for Quantum Query Lower Bounds

TL;DR

This work extends quantum lower-bound techniques to the random permutation model by introducing a twirled, strictly monotone-factorized permutation oracle (TSPO) and a corresponding Fundamental Lemma. It defines a robust progress-measure framework to bound the probability that a quantum algorithm finds an input-output pair satisfying a relation, culminating in a general bound for queries. The authors apply this to the one-round sponge, proving unconditional preimage resistance in the random permutation model, and to Unruh’s double-sided zero-search conjecture, providing explicit polynomial-query upper bounds. The results offer a versatile tool for quantum query complexity in permutation settings and set the stage for further generalizations in cryptographic security analyses under quantum models.

Abstract

We propose a generalization of Zhandry's compressed oracle method to random permutations, where an algorithm can query both the permutation and its inverse. We show how to use the resulting oracle simulation to bound the success probability of an algorithm for any predicate on input-output pairs, a key feature of Zhandry's technique that had hitherto resisted attempts at generalization to random permutations. One key technical ingredient is to use strictly monotone factorizations to represent the permutation in the oracle's database. As an application of our framework, we show that the one-round sponge construction is unconditionally preimage resistant in the random permutation model. This proves a conjecture by Unruh.
Paper Structure (29 sections, 32 theorems, 131 equations, 1 figure)

This paper contains 29 sections, 32 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Random Permutation Model.
  3. Summary of Contributions
  4. Key Challenges and Techniques
  5. Random Permutations from Independent Transpositions.
  6. The Permutation Oracle.
  7. A Fundamental Lemma for Permutation Oracles.
  8. Twirling the Oracle.
  9. The Progress Measure.
  10. Sparsity Analysis.
  11. Putting Things Together.
  12. Concurrent Independent Work
  13. Preliminaries
  14. Combinatorics.
  15. Quantum Information.
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\mathcal{A}$ be an algorithm with quantum query access to a random permutation $\pi\in S_N$ and its inverse $\pi^{-1}$, and let $R$ be a relation. If $\mathcal{A}$ makes at most $q$ queries and outputs $x$, then where $r_{\max} = \max \lbrace*\rbrace{ \max_x \, \lvert R_x\rvert, \max_y \, \lvert R^\mathrm{inv}_y\rvert }$, with $R_x = \{ y : (x,y) \in R \}$ and $R^\mathrm{inv}_y = \{ x : (x,y

Figures (1)

  • Figure 1: The 1-round sponge.

Theorems & Definitions (63)

  • Theorem 1.1: Informal
  • Corollary 1.2: One-Round Sponge, informal
  • Corollary 1.3: Double-Sided Zero-Search, informal
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4: Active sets
  • Lemma 3.5
  • ...and 53 more