International Association
for Cryptologic Research

Two fundamental properties of quantum states that quantum information theory explores are pseudorandomness and provability of destruction. We introduce the notion of quantum pseudorandom states with proofs of destruction (PRSPD) that combines both these properties. Like standard pseudorandom states (PRS), these are efficiently generated quantum states that are indistinguishable from random, but they can also be measured to create a classical string. This string is verifiable (given the secret key) and certifies that the state has been destructed. We show that, similarly to PRS, PRSPD can be constructed from any post-quantum one-way function. As far as the authors are aware, this is the first construction of a family of states that satisfies both pseudorandomness and provability of destruction. We show that many cryptographic applications that were shown based on PRS variants using quantum communication can be based on (variants of) PRSPD using only classical communication. This includes symmetric encryption, message authentication, one-time signatures, commitments, and classically verifiable private quantum coins.

In the framework of Impagliazzo's five worlds, a distinction is often made between two worlds, one where public-key encryption exists (Cryptomania), and one in which only one-way functions exist (MiniCrypt). However, the boundaries between these worlds can change when quantum information is taken into account. Recent work has shown that quantum variants of oblivious transfer and multi-party computation, both primitives that are classically in Cryptomania, can be constructed from one-way functions, placing them in the realm of quantum MiniCrypt (the so-called MiniQCrypt). This naturally raises the following question: Is it possible to construct a quantum variant of public-key encryption, which is at the heart of Cryptomania, from one-way functions or potentially weaker assumptions? In this work, we initiate the formal study of the notion of quantum public-key encryption (qPKE), i.e., public-key encryption where keys are allowed to be quantum states. We propose new definitions of security and several constructions of qPKE based on the existence of one-way functions (OWF), or even weaker assumptions, such as pseudorandom function-like states (PRFS) and pseudorandom function-like states with proof of destruction (PRFSPD). Finally, to give a tight characterization of this primitive, we show that computational assumptions are necessary to build quantum public-key encryption. That is, we give a self-contained proof that no quantum public-key encryption scheme can provide information-theoretic security.

Quantum money allows a bank to mint quantum money states that can later be verified and cannot be forged. Usually, this requires a quantum communication infrastructure to perform transactions. Gavinsky (CCC 2012) introduced the notion of classically verifiable quantum money, which allows verification through classical communication. In this work, we introduce the notion of classical minting and combine it with classical verification to introduce semi-quantum money. Semi-quantum money is the first type of quantum money to allow transactions with completely classical communication and an entirely classical bank. This work features constructions for both a public memory-dependent semi-quantum money scheme and a private memoryless semi-quantum money scheme. The public construction is based on the works of Zhandry  and Coladangelo, and the private construction is based on the notion of noisy trapdoor claw-free functions (NTCF) introduced by Brakerski et al. (FOCS 2018). In terms of technique, our main contribution is a perfect parallel repetition theorem for NTCF.

In (single-server) Private Information Retrieval (PIR), a server holds a large database $${\mathtt {DB}}$$ of size n, and a client holds an index $$i \in [n]$$ and wishes to retrieve $${\mathtt {DB}}[i]$$ without revealing i to the server. It is well known that information theoretic privacy even against an “honest but curious” server requires $$\varOmega (n)$$ communication complexity. This is true even if quantum communication is allowed and is due to the ability of such an adversarial server to execute the protocol on a superposition of databases instead of on a specific database (“input purification attack”).Nevertheless, there have been some proposals of protocols that achieve sub-linear communication and appear to provide some notion of privacy. Most notably, a protocol due to Le Gall (ToC 2012) with communication complexity $$O(\sqrt{n})$$ , and a protocol by Kerenidis et al. (QIC 2016) with communication complexity $$O(\log (n))$$ , and O(n) shared entanglement.We show that, in a sense, input purification is the only potent adversarial strategy, and protocols such as the two protocols above are secure in a restricted variant of the quantum honest but curious (a.k.a specious) model. More explicitly, we propose a restricted privacy notion called anchored privacy, where the adversary is forced to execute on a classical database (i.e. the execution is anchored to a classical database). We show that for measurement-free protocols, anchored security against honest adversarial servers implies anchored privacy even against specious adversaries.Finally, we prove that even with (unlimited) pre-shared entanglement it is impossible to achieve security in the standard specious model with sub-linear communication, thus further substantiating the necessity of our relaxation. This lower bound may be of independent interest (in particular recalling that PIR is a special case of Fully Homomorphic Encryption).