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Cryptography from Pseudorandom Quantum States

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13507))

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Abstract

Pseudorandom states, introduced by Ji, Liu and Song (Crypto’18), are efficiently-computable quantum states that are computationally indistinguishable from Haar-random states. One-way functions imply the existence of pseudorandom states, but Kretschmer (TQC’20) recently constructed an oracle relative to which there are no one-way functions but pseudorandom states still exist. Motivated by this, we study the intriguing possibility of basing interesting cryptographic tasks on pseudorandom states.

We construct, assuming the existence of pseudorandom state generators that map a \(\lambda \)-bit seed to a \(\omega (\log \lambda )\)-qubit state, (a) statistically binding and computationally hiding commitments and (b) pseudo one-time encryption schemes. A consequence of (a) is that pseudorandom states are sufficient to construct maliciously secure multiparty computation protocols in the dishonest majority setting.

Our constructions are derived via a new notion called pseudorandom function-like states (PRFS), a generalization of pseudorandom states that parallels the classical notion of pseudorandom functions. Beyond the above two applications, we believe our notion can effectively replace pseudorandom functions in many other cryptographic applications.

H. Yuen—Supported by AFOSR award FA9550-21-1-0040 and NSF CAREER award CCF-2144219. L. Qian—Supported by DARPA under Agreement No. HR00112020023.

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Notes

  1. 1.

    Both the works [4, 19] explicitly raised the question of basing secure computation on assumptions weaker than one-way functions.

  2. 2.

    Recall that \(\lambda \) is the key length.

  3. 3.

    In the technical sections, we define a QPT algorithm \(\textsf{Test}\) that given a state \(\rho \) along with k, x, determines if \(\rho \) is equal to the output G(k, x). We show the existence of such a test algorithm for any PRFS.

  4. 4.

    To simplify the analysis, there is an additional technical property of the PRFS not mentioned here that is required by our construction, called recognizable abort (Definition 4). All known constructions of PRFS and PRS (including ours) have the recognizable abort property.

  5. 5.

    However, in an updated draft of [26], the authors sketch how, for a special form of quantum commitment schemes, sum-binding does imply our notion of statistical binding.

  6. 6.

    The sum of probabilities that an adversarial decommitter can decommit to 0 and to 1 in the ideal world of our definition (Definition 6) and therefore they sum up to at most negligibly larger than 1 in the real world by our statistical binding guarantee.

  7. 7.

    The majority of the authors of this paper believe one-way functions exist.

  8. 8.

    One can think of \({\left| {\bot } \right\rangle } \) as the \((n + 1)\)-qubit state \({\left| {100\cdots 0} \right\rangle } \) with the first qubit indicating whether the generator aborted or not. If the generator doesn’t abort, then it outputs \({\left| {0} \right\rangle } \otimes {\left| {\psi } \right\rangle } \) for some pure state \({\left| {\psi } \right\rangle } \) (called the correct output state of G on input (k, x)). The distinguisher in the definition of PRFS generator would then only get the last n qubits as input.

  9. 9.

    The argument is as follows: if \(\eta \) were on average noticeably far from 1, then a purity test using SWAP tests would distinguish the outputs from Haar random states which are pure. This is formalized in the full version.

  10. 10.

    See the full version for a definition of the unitary part of a generalized quantum circuit.

  11. 11.

    To sample \(P = \bigotimes _i P_i\), the receiver can sample uniformly random bits \(\alpha _1,\beta _1,\ldots ,\alpha _m,\beta _m\), and let \(P_i = X^{\alpha _i} Z^{\beta _i}\) where X and Z are the single-qubit Pauli operators.

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Acknowledgements

We thank Tomoyuki Morimae, Takashi Yamakawa, Jun Yan, and Fermi Ma for their very helpful feedback and discussions about pseudorandom quantum states.

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Authors and Affiliations

  1. UCSB, Santa Barbara, USA

    Prabhanjan Ananth

  2. Boston University, Boston, USA

    Luowen Qian

  3. Columbia University, New York, USA

    Henry Yuen

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Correspondence to Luowen Qian .

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  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Ananth, P., Qian, L., Yuen, H. (2022). Cryptography from Pseudorandom Quantum States. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13507. Springer, Cham. https://doi.org/10.1007/978-3-031-15802-5_8

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  • DOI: https://doi.org/10.1007/978-3-031-15802-5_8

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