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Quantum Time-Space Tradeoff for Finding Multiple Collision Pairs

Authors Yassine Hamoudi , Frédéric Magniez

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Author Details

Yassine Hamoudi
  • Université de Paris, IRIF, CNRS, F-75013 Paris, France
Frédéric Magniez
  • Université de Paris, IRIF, CNRS, F-75013 Paris, France

Funding

This research was supported in part by the ERA-NET Cofund in Quantum Technologies project QuantAlgo and the French ANR Blanc project RDAM.

Acknowledgements

The authors want to thank the anonymous referees for their valuable comments and suggestions which helped to improve this paper.

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Yassine Hamoudi and Frédéric Magniez. Quantum Time-Space Tradeoff for Finding Multiple Collision Pairs. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 1:1-1:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.TQC.2021.1

Abstract

We study the problem of finding K collision pairs in a random function f : [N] → [N] by using a quantum computer. We prove that the number of queries to the function in the quantum random oracle model must increase significantly when the size of the available memory is limited. Namely, we demonstrate that any algorithm using S qubits of memory must perform a number T of queries that satisfies the tradeoff T³ S ≥ Ω(K³N). Classically, the same question has only been settled recently by Dinur [Dinur, 2020], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener [Oorschot and Wiener, 1999] achieves the optimal time-space tradeoff of T² S = Θ(K² N). Our result limits the extent to which quantum computing may decrease this tradeoff. Our method is based on a novel application of Zhandry’s recording query technique [Zhandry, 2019] for proving lower bounds in the exponentially small success probability regime. As a second application, we give a simpler proof of the time-space tradeoff T² S ≥ Ω(N³) for sorting N numbers on a quantum computer, which was first obtained by Klauck, Špalek and de Wolf [Klauck et al., 2007].

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum computing
  • query complexity
  • lower bound
  • time-space tradeoff

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