On the Computational Hardness Needed for Quantum Cryptography

Authors Zvika Brakerski , Ran Canetti , Luowen Qian

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Zvika Brakerski
  • Weizmann Institute of Science, Rehovot, Israel
Ran Canetti
  • Boston University, MA, US
Luowen Qian
  • Boston University, MA, US


LQ thanks Prabhanjan Ananth and Henry Yuen for the motivating discussions prior to this work. LQ is also grateful to Scott Aaronson and William Kretschmer for their insightful discussions for Section 1.2. Some of these discussions occurred while LQ was visiting the Simons Institute for the Theory of Computing. We thank Fermi Ma for his helpful comments on a preliminary version of this work.

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Zvika Brakerski, Ran Canetti, and Luowen Qian. On the Computational Hardness Needed for Quantum Cryptography. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 24:1-24:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In the classical model of computation, it is well established that one-way functions (OWF) are minimal for computational cryptography: They are essential for almost any cryptographic application that cannot be realized with respect to computationally unbounded adversaries. In the quantum setting, however, OWFs appear not to be essential (Kretschmer 2021; Ananth et al., Morimae and Yamakawa 2022), and the question of whether such a minimal primitive exists remains open. We consider EFI pairs - efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quantum states. Building on the work of Yan (2022), which shows equivalence between EFI pairs and statistical commitment schemes, we show that EFI pairs are necessary for a large class of quantum-cryptographic applications. Specifically, we construct EFI pairs from minimalistic versions of commitments schemes, oblivious transfer, and general secure multiparty computation, as well as from QCZK proofs from essentially any non-trivial language. We also construct quantum computational zero knowledge (QCZK) proofs for all of QIP from any EFI pair. This suggests that, for much of quantum cryptography, EFI pairs play a similar role to that played by OWFs in the classical setting: they are simple to describe, essential, and also serve as a linchpin for demonstrating equivalence between primitives.

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
  • Theory of computation → Cryptographic protocols
  • Theory of computation → Computational complexity and cryptography
  • Security and privacy → Mathematical foundations of cryptography
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Quantum complexity theory
  • quantum cryptography
  • efi
  • commitment scheme
  • oblivious transfer
  • zero knowledge
  • secure multiparty computation


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