Our main result is a quantum polynomial-time reduction from the group action discrete logarithm (DLP) problem to a specific cloning problem. A consequence of this result is that the public-key quantum money scheme proposed by Zhandry (2024), which is based on abelian group actions, is secure in the generic group action model. Specifically, our result shows that breaking the quantum money scheme is equivalent, under quantum polynomial-time reductions, to solving the group action DLP. An immediate implication of our result concerns the relationship between cloning and preparing Fourier states: our main theorem shows that the problem of cloning group action Fourier states is equivalent to the problem of preparing them.
Our main result is a quantum polynomial-time reduction from the group action discrete logarithm (DLP) problem to a specific cloning problem. A consequence of this result is that the public-key quantum money scheme proposed by Zhandry (2024), which is based on abelian group actions, is secure in the generic group action model. Specifically, our result shows that breaking the quantum money scheme is equivalent, under quantum polynomial-time reductions, to solving the group action DLP. An immediate implication of our result concerns the relationship between cloning and preparing Fourier states: our main theorem shows that the problem of cloning group action Fourier states is equivalent to the problem of preparing them.
We give a construction of public key quantum money, and even a strengthened version called quantum lightning, from abelian group actions, which can in turn be constructed from suitable isogenies over elliptic curves. We prove security in the generic group model for group actions under a plausible computational assumption, and develop a general toolkit for proving quantum security in this model. Along the way, we explore knowledge assumptions and algebraic group actions in the quantum setting, finding significant limitations of these assumptions/models compared to generic group actions. 62 pages. This is the TheoretiCS journal version
We give a construction of public key quantum money, and even a strengthened version called quantum lightning, from abelian group actions, which can in turn be constructed from suitable isogenies over elliptic curves. We prove security in the generic group model for group actions under a plausible computational assumption, and develop a general toolkit for proving quantum security in this model. Along the way, we explore knowledge assumptions and algebraic group actions in the quantum setting, finding significant limitations of these assumptions/models compared to generic group actions.
Aaronson, Atia, and Susskind (2020) established that efficiently mapping between quantum states ∣ψ⟩ and ∣ϕ⟩ is computationally equivalent to distinguishing their superpositions ∣ψ⟩±∣ϕ⟩. We generalize this insight into a broader duality principle, wherein manipulating quantum states in one basis is equivalent to extracting their value in a complementary basis. This general duality principle states that the ability to implement a unitary representation of a group is computationally equivalent to the ability to perform a Fourier subspace extraction from its irreducible representations. Building on our duality principle, we present the following applications: * We extend the construction of publicly-key quantum money of Zhandry (2024) from Abelian group actions to a construction of quantum lightning from non-Abelian group actions, and eliminate Zhandry's reliance on a black-box model for justifying security. Instead, we prove a direct reduction to a computational assumption -- the pre-action security of cryptographic group actions. Our construction is realizable with symmetric group actions, including those implicit in the McEliece cryptosystem. * We provide an alternative quantum lightning construction from one-way homomorphisms, with security holding under certain conditions. This scheme shows equivalence among four security notions: quantum lightning security, worst-case and average-case cloning security, and security against preparing a canonical state. * We formalize the notion of quantum fire, states that are efficiently clonable, but not efficiently telegraphable. These states can be spread like fire, provided they are kept alive quantumly and do not decohere. The only previously known construction relied on a unitary quantum oracle, whereas we present the first candidate construction of quantum fire using a classical oracle.
Public verification of quantum money has been one of the central objects in quantum cryptography ever since Wiesner's pioneering idea of using quantum mechanics to construct banknotes against counterfeiting. So far, we do not know any publicly-verifiable quantum money scheme that is provably secure from standard assumptions. In this work, we provide both negative and positive results for publicly verifiable quantum money. **In the first part, we give a general theorem, showing that a certain natural class of quantum money schemes from lattices cannot be secure. We use this theorem to break the recent quantum money scheme of Khesin, Lu, and Shor. **In the second part, we propose a framework for building quantum money and quantum lightning we call invariant money which abstracts some of the ideas of quantum money from knots by Farhi et al.(ITCS'12). In addition to formalizing this framework, we provide concrete hard computational problems loosely inspired by classical knowledge-of-exponent assumptions, whose hardness would imply the security of quantum lightning, a strengthening of quantum money where not even the bank can duplicate banknotes. **We discuss potential instantiations of our framework, including an oracle construction using cryptographic group actions and instantiations from rerandomizable functional encryption, isogenies over elliptic curves, and knots.
Public-key quantum money is a cryptographic protocol in which a bank can create quantum states which anyone can verify but no one except possibly the bank can clone or forge. There are no secure public-key quantum money schemes in the literature; as we show in this paper, the only previously published scheme [1] is insecure. We introduce a category of quantum money protocols which we call collision-free. For these protocols, even the bank cannot prepare multiple identical-looking pieces of quantum money. We present a blueprint for how such a protocol might work as well as a concrete example which we believe may be insecure.
The security of quantum money from knots, and of its generalization to invariant money, is based on the assumption that path-finding, exhibiting a sequence of moves between two equivalent objects, is hard. No proof of security from that assumption alone is known. The existing proofs add knowledge-of-path assumptions, which assert that any efficient algorithm producing two objects with the same invariant implicitly knows a path between them. No attack can refute such an assumption, and it is not known to follow from security. We ask when path-finding is the right assumption. When each equivalence class is the orbit of an efficiently computable action of a group that can be superposed over, and every move acts as a group element, as for graphs, average-case hardness of path-finding is necessary for security. For knots no such group is known, and a path-finder only reduces forgery to an equally hard state-preparation problem. With or without a path-finder, a forger must prepare a state that verification accepts, and we take the hardness of that task as the assumption. For schemes whose verification walk mixes in polynomial time, the preparation assumption states that no efficient algorithm, given the serial number of a freshly minted banknote and one object measured from it, prepares such a state. It is falsifiable, and it is equivalent to security against forgers that measure their banknote first. The transfer assumption, which security implies, states that measuring first costs a forger at most a polynomial factor. Together the two are equivalent to security, so every proof of security must establish the preparation assumption. If the preparation assumption holds, no fully black-box reduction that calls the forger only at the serial number it is given can derive the transfer assumption from the preparation assumption.