Hard Quantum Extrapolations in Quantum Cryptography

TL;DR

This paper investigates the quantum analogues of classical extrapolation tasks, establishing their complexity and connections to fundamental quantum cryptographic primitives, thereby advancing understanding of minimal assumptions for quantum cryptography.

Contribution

It introduces the classical→quantum extrapolation task, links its hardness to quantum cryptographic primitives, and proposes a fully quantum generalization.

Findings

Classical→quantum extrapolation is hard if quantum commitments exist.

It is easy in quantum polynomial space.

Connections are established between extrapolation hardness and quantum cryptographic primitives.

Abstract

Although one-way functions are well-established as the minimal primitive for classical cryptography, a minimal primitive for quantum cryptography is still unclear. Universal extrapolation, first considered by Impagliazzo and Levin (1990), is hard if and only if one-way functions exist. Towards better understanding minimal assumptions for quantum cryptography, we study the quantum analogues of the universal extrapolation task. Specifically, we put forth the classical$\rightarrow$quantum extrapolation task, where we ask to extrapolate the rest of a bipartite pure state given the first register measured in the computational basis. We then use it as a key component to establish new connections in quantum cryptography: (a) quantum commitments exist if classical$\rightarrow$quantum extrapolation is hard; and (b) classical$\rightarrow$quantum extrapolation is hard if any of the following cryptographic primitives exists: quantum public-key cryptography (such as quantum money and signatures) with a classical public key or 2-message quantum key distribution protocols. For future work, we further generalize the extrapolation task and propose a fully quantum analogue. We show that it is hard if quantum commitments exist, and it is easy for quantum polynomial space.