Quantum time/memory/data tradeoff attacks

Orr Dunkelman, Nathan Keller, Eyal Ronen, Adi Shamir

Research output: Contribution to journalArticlepeer-review


One of the most celebrated and useful cryptanalytic algorithms is Hellman’s time/memory tradeoff (and its Rainbow Table variant), which can be used to invert random-looking functions with domains of size N with time and space complexities satisfying TM2= N2 . In this paper we develop new upper bounds on their performance in the quantum setting. As a search problem, one can always apply to it the standard Grover’s algorithm, but this algorithm does not benefit from the possible availability of a large memory in which one can store auxiliary advice obtained during a free preprocessing stage. In fact, at FOCS’20 it was rigorously shown that for memory size bounded by M≤O(N) , even quantum advice cannot yield an attack which is better than Grover’s algorithm.Our main result complements this lower bound by showing that in the standard Quantum Accessible Classical Memory (QACM) model of computation, we can improve Hellman’s tradeoff curve to T4 / 3M2= N2 . When we generalize the cryptanalytic problem to a time/memory/data tradeoff attack (in which one has to invert f for at least one of D given values), we get the generalized curve T4 / 3M2D2= N2 . A typical point on this curve is D= N0.2 , M= N0.6 , and T= N0.3 , whose time is strictly lower than both Grover’s algorithm (which requires T= N0.4 in this generalized search variant) and the classical Hellman algorithm (which requires T= N0.4 for these D and M).

Original languageEnglish
Pages (from-to)159-177
Number of pages19
JournalDesigns, Codes, and Cryptography
Issue number1
StatePublished - Jan 2024

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


  • Hellman tables
  • Quantum cryptanalysis
  • Rainbow tables
  • TMD attacks

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science Applications
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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