Abstract
Pseudorandom quantum states (PRS) are efficiently constructible states that are computationally indistinguishable from being Haar-random, and have recently found cryptographic applications. We explore new definitions, new properties and applications of pseudorandom states, and present the following contributions:
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1.
New Definitions: We study variants of pseudorandom function-like state (PRFS) generators, introduced by Ananth, Qian, and Yuen (CRYPTO’22), where the pseudorandomness property holds even when the generator can be queried adaptively or in superposition. We show feasibility of these variants assuming the existence of post-quantum one-way functions.
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2.
Classical Communication: We show that PRS generators with logarithmic output length imply commitment and encryption schemes with classical communication. Previous constructions of such schemes from PRS generators required quantum communication.
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Simplified Proof: We give a simpler proof of the Brakerski–Shmueli (TCC’19) result that polynomially-many copies of uniform superposition states with random binary phases are indistinguishable from Haar-random states.
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4.
Necessity of Computational Assumptions: We also show that a secure PRS with output length logarithmic, or larger, in the key length necessarily requires computational assumptions.
L. Qian: Supported by DARPA under Agreement No. HR00112020023.
H. Yuen: Supported by AFOSR award FA9550-21-1-0040 and NSF CAREER award CCF-2144219.
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Notes
- 1.
However, unlike the equivalence between PRG and PRF in the classical setting [8], it is not known whether every PRFS generator can be constructed from PRS generators in a black-box way.
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For example, the application of private-key encryption from PRFS as described in [2] is only selectively secure. This is due to the fact that the underlying PRFS is selectively secure.
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We also note that there is a much more roundabout argument for a quantitatively weaker result: [2] constructed bit commitment schemes from \(O(\log \lambda )\)-length PRS. If such PRS were possible to construct unconditionally, this would imply information-theoretically secure bit commitment schemes in the quantum setting. However, this contradicts the famous results of [13, 15], which rules out this possibility. Our calculation, on the other hand, directly shows that \(\log \lambda \) (without any constants in front) is a sharp threshold.
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A density matrix \(\rho \) has purity p if \(\textrm{Tr}(\rho ^2)=p\).
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This in turn can be built from \(O(\log (\lambda ))\)-output PRS as shown in [2].
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Alternatively, one can think of answer registers \(\textbf{Y}_1,\textbf{Y}_2,\ldots \) as being initialized in the zeroes state at the beginning, and the query algorithm is only allowed to act nontrivially on \(\textbf{Y}_i\) after the i’th query.
- 8.
Alternatively, one can think of the oracle as an isometry mapping register \(\textbf{X}\) to registers \(\textbf{X} \textbf{Y}\).
- 9.
It is stronger in the sense that an algorithm that has quantum query access to the oracle can simulate an algorithm that only has classical query access.
- 10.
In this illustration, we are pretending that the PRFS satisfies perfect state generation property. That is, the output of PRFS is always a pure state.
- 11.
For readers familiar with [12], it can be verified that a sufficient condition for that proof to go through is if \(2^\lambda \cdot e^{-2^n/3}\) is negligible, which is satisfied if \(n \ge \log \lambda + 2\).
References
Ambainis, A., Emerson, J.: Quantum t-designs: t-wise independence in the quantum world. In: 22nd Annual IEEE Conference on Computational Complexity (CCC 2007), 13–16 June 2007, San Diego, California, USA, pp. 129–140. IEEE Computer Society (2007)
Ananth, P., Qian, L., Yuen, H.: Cryptography from pseudorandom quantum states. In: Annual International Cryptology Conference 2022, pp. 208–236. Springer, Cham (2022)
Brakerski, Z., Shmueli, O.: (Pseudo) random quantum states with binary phase. In: Hofheinz, D., Rosen, A. (eds.) TCC 2019. LNCS, vol. 11891, pp. 229–250. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-36030-6_10
Brakerski, Z., Shmueli, O.: Scalable pseudorandom quantum states. In: Micciancio, D., Ristenpart, T., (eds.), Advances in Cryptology - CRYPTO 2020–40th Annual International Cryptology Conference, CRYPTO 2020, Santa Barbara, CA, USA, August 17–21, 2020, Proceedings, Part II LNCS, vol. 12171, pp. 417–440. Springer (2020)
Brandão, F.G.S.L., Harrow, A.W., Horodecki, M.: Local random quantum circuits are approximate polynomial-designs. Commun. Math. Phys. 346(2), 397–434 (2016). https://doi.org/10.1007/s00220-016-2706-8
Dankert, C., Cleve, R., Emerson, J., Livine, E.: Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. 80, 012304 (2009)
Gavinsky, D.: Quantum money with classical verification. In: Proceedings of the 27th Conference on Computational Complexity, CCC 2012, Porto, Portugal, June 26–29, 2012, pp. 42–52. IEEE Computer Society (2012)
Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)
Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)
Huang, H.Y., Kueng, R., Preskill, J.: Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16(10), 1050–1057 (2020)
Ji, Z., Liu, Y.-K., Song, F.: Pseudorandom quantum states. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10993, pp. 126–152. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_5
Kretschmer, W.: Quantum pseudorandomness and classical complexity. In Hsieh, M.H., (ed.) 16th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2021, July 5–8, 2021, Virtual Conference, vol. 197 of LIPIcs, pp. 2:1–2:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
Lo, H.K., Chau, H.F.: Is quantum bit commitment really possible? Phys. Rev. Lett. 78, 3410–3413 (1997)
Lowe, A.: Learning quantum states without entangled measurements. Master’s thesis (2021)
Mayers, D.: Unconditionally secure quantum bit commitment is impossible. Phys. Rev. Lett. 78, 3414–3417 (1997)
Morimae, T., Yamakawa, T.: Quantum commitments and signatures without one-way functions. In: CRYPTO (2022)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press (2010)
Roy, A., Scott, A.J.: Unitary designs and codes. Des. Codes Cryptography, 53(1), 13–31 (2009)
Zhandry, M.: How to construct quantum random functions. In: 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20–23, 2012, pp. 679–687. IEEE Computer Society (2012)
Zhandry, M.: Secure identity-based encryption in the quantum random oracle model. In: Reihaneh, S.-N., Canetti, R., (eds.), Advances in Cryptology - CRYPTO 2012–32nd Annual Cryptology Conference, Santa Barbara, CA, USA, 19–23 August 2012. Proceedings, LNCS, vol. 7417, pp. 758–775. Springer (2012)
Acknowledgements
The authors would like to thank the anonymous TCC 2022 reviewers for their helpful comments. The authors would also like to thank Fermi Ma for his suggestions that improved the bounds and the analysis in the proof of binary phase PRS.
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Ananth, P., Gulati, A., Qian, L., Yuen, H. (2022). Pseudorandom (Function-Like) Quantum State Generators: New Definitions and Applications. In: Kiltz, E., Vaikuntanathan, V. (eds) Theory of Cryptography. TCC 2022. Lecture Notes in Computer Science, vol 13747. Springer, Cham. https://doi.org/10.1007/978-3-031-22318-1_9
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