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2025 | OriginalPaper | Chapter

Real-Valued Somewhat-Pseudorandom Unitaries

Authors : Zvika Brakerski, Nir Magrafta

Published in: Theory of Cryptography

Publisher: Springer Nature Switzerland

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Abstract

We explore a very simple distribution of unitaries: random (binary) phase—Hadamard—random (binary) phase—random computational-basis permutation. We show that this distribution is statistically indistinguishable from random Haar unitaries for any polynomial set of orthogonal input states (in any basis) with polynomial multiplicity. This shows that even though real-valued unitaries cannot be completely pseudorandom (Haug, Bharti, Koh, arXiv:2306.11677), we can still obtain some pseudorandom properties without giving up on the simplicity of a real-valued unitary.

Our analysis shows that an even simpler construction: applying a random (binary) phase followed by a random computational-basis permutation, would suffice, assuming that the input is orthogonal and flat (that is, has high min-entropy when measured in the computational basis).

Using quantum-secure one-way functions (which imply quantum-secure pseudorandom functions and permutations), we obtain an efficient cryptographic instantiation of the above.

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previous chapter Quantum Pseudorandom Scramblers

next chapter Split-State Non-malleable Codes and Secret Sharing Schemes for Quantum Messages

In a formal definition, one has to address the subtlety of whether “efficiency” is defined with respect to the input length n, or with respect to some “security parameter”. This distinction does not matter for our current discussion, and we will point out when it does down the line.

In [15] there is an additional construction which uses complex unitaries, but for the PRSS property, a real valued construction suffices.

We notice that \(k=1\) is not possible for reasons of symmetry and therefore it does not appear in the sum, but we could have achieved our result even without this minor optimization.

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Title: Real-Valued Somewhat-Pseudorandom Unitaries
Authors: Zvika Brakerski
Nir Magrafta
Publisher: Springer Nature Switzerland
Book: Theory of Cryptography
Print ISBN: 978-3-031-78016-5

Electronic ISBN: 978-3-031-78017-2

Copyright Year: 2025
DOI: https://doi.org/10.1007/978-3-031-78017-2_2

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