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 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.
Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is (1) public-key, meaning that anyone can verify a banknote as genuine, not only the bank that printed it, and (2) cryptographically secure, under a "classical" hardness assumption that has nothing to do with quantum money. Our scheme is based on hidden subspaces, encoded as the zero-sets of random multivariate polynomials. A main technical advance is to show that the "black-box" version of our scheme, where the polynomials are replaced by classical oracles, is unconditionally secure. Previously, such a result had only been known relative to a quantum oracle (and even there, the proof was never published). Even in Wiesner's original setting -- quantum money that can only be verified by the bank -- we are able to use our techniques to patch a major security hole in Wiesner's scheme. We give the first private-key quantum money scheme that allows unlimited verifications and that remains unconditionally secure, even if the counterfeiter can interact adaptively with the bank. Our money scheme is simpler than previous public-key quantum money schemes, including a knot-based scheme of Farhi et al. The verifier needs to perform only two tests, one in the standard basis and one in the Hadamard basis -- matching the original intuition for quantum money, based on the existence of complementary observables. Our security proofs use a new variant of Ambainis's quantum adversary method, and several other tools that might be of independent interest.
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.
Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner's scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copy-protected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publicly-verifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its input-output behavior can be quantumly copy-protected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a "Complexity-Theoretic No-Cloning Theorem," which generalizes both the standard No-Cloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum t-designs. Moving beyond the oracle world, we also present an explicit candidate scheme for publicly-verifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copy-protecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption).
Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is (1) public-key---meaning that anyone can verify a banknote as genuine, not only the bank that printed it, and (2) cryptographically secure, under a "classical" hardness assumption that has nothing to do with quantum money. Our scheme is based on hidden subspaces, encoded as the zero-sets of random multivariate polynomials. A main technical advance is to show that the "black-box" version of our scheme, where the polynomials are replaced by classical oracles, is unconditionally secure. Previously, such a result had only been known relative to a quantum oracle (and even there, the proof was never published).
Quantum information allows us to build quantum money schemes, where a bank can issue banknotes in the form of authenticatable quantum states that cannot be cloned or counterfeited. Similar to paper banknotes, in existing quantum money schemes, a banknote consists of an unclonable quantum state and a classical serial number, signed by bank. Thus, they lack one of the most fundamental properties cryptographers look for in a currency scheme: privacy. In this work, we first further develop the formal definitions of privacy for quantum money schemes. Then, we construct the first public-key quantum money schemes that satisfy these security notions. Namely, - Assuming existence of indistinguishability obfuscation (iO) and hardness of Learning with Errors (LWE), we construct a public-key quantum money scheme with anonymity against users and traceability by authorities. Since it is a policy choice whether authorities should be able to track banknotes or not, we also construct an untraceable money scheme from the same cryptographic assumptions, where no one (not even the authorities) can track banknotes. Further, we show that the no-cloning principle, a result of quantum mechanics, allows us to construct schemes, with security guarantees that are classically impossible, for a seemingly unrelated application: voting! - Assuming iO and LWE, we construct a universally verifiable quantum voting scheme with classical votes. Finally, as a technical tool, we introduce the notion of publicly rerandomizable encryption with strong correctness, where no adversary is able to produce a malicious ciphertext and a malicious randomness such that the ciphertext before and after rerandomization decrypts to different values! We believe this might be of independent interest. - Assuming LWE, we construct a (post-quantum) classical publicly rerandomizable encryption scheme with strong correctness.