Quantum money is the task of verifying the validity of banknotes while ensuring that they cannot be counterfeited. Public-key quantum money allows anyone to perform verification, while the private-key setting restricts the ability to verify to banks, as in Wiesner's original scheme. The current state of technological progress means that errors are impossible to entirely suppress, hence the requirement for noise-tolerant schemes. We show for the first time how to achieve noise-tolerance in the public-key setting. Our techniques follow Aaronson and Christiano's oracle model, where we use the ideas of quantum error correction to extend their scheme: a valid banknote is now a subspace state possibly affected by noise, and verification is performed by using classical oracles to check for membership in "larger spaces." Additionally, a banknote in our scheme is minted by preparing conjugate coding states and applying a unitary that permutes the standard basis vectors.
Quantum money is the task of verifying the validity of banknotes while ensuring that they cannot be counterfeited. Public-key quantum money allows anyone to perform verification, while the private-key setting restricts the ability to verify to banks, as in Wiesner's original scheme. The current state of technological progress means that errors are impossible to entirely suppress, hence the requirement for noise-tolerant schemes. We show for the first time how to achieve noise-tolerance in the public-key setting. Our techniques follow Aaronson and Christiano's oracle model, where we use the ideas of quantum error correction to extend their scheme: a valid banknote is now a subspace state possibly affected by noise, and verification is performed by using classical oracles to check for membership in "larger spaces." Additionally, a banknote in our scheme is minted by preparing conjugate coding states and applying a unitary that permutes the standard basis vectors.
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.
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).
There are various practical obstacles in quantum cryptography. These include the problems that arise in implementing quantum cryptographic protocols with current and near-term technology, and those that come with the development and security analysis of new protocols. In this work, we examine a few of these obstacles and, for each one, we highlight an approach that aims to mitigate it. We start by examining the setting of device-independence, with oblivious transfer in the bounded-quantum-storage-model as the primitive of interest. The typical approach to enabling device-independence is the use of a Bell-inequality violation, which requires a strict non-communication assumption that is practically difficult to enforce. We develop and analyze the security of a device-independent oblivious transfer protocol that replaces the Bell-inequality violation approach with a computational assumption, thereby relaxing the non-communication assumption. Our second topic examines the problem of noise in a quantum money scheme. We show how to achieve noise-tolerance in the public-key setting. Our scheme can be seen as a generalization of the [AC12] scheme: a valid banknote is now a subspace state possibly affected by noise, and verification is performed by using classical oracles to check for membership in "larger spaces." Additionally, a banknote in our scheme is minted by preparing conjugate coding states and applying a unitary. For our last topic, we consider the general task of proving security against a specious adversary, which is the quantum analogue of the classical semi-honest adversary. A useful feature of specious adversaries is that security against them can guarantee security against stronger adversaries [DNS12]. This motivates the search for a canonical specious adversary, where proving security against such an adversary guarantees security against the entire class of specious adversaries. Within a restricted setting, we define a particular specious adversary and show that it satisfies our definition of canonical.
We introduce a private quantum money scheme with the note verification procedure based on sampling matching, a problem in a one-way communication complexity model. Our scheme involves a bank who produces and distributes quantum notes, noteholders who are untrusted, and trusted local verifiers of the bank to whom the holders send their notes in order to carry out transactions. The key aspects of our money scheme include: note verification procedure requiring a single round classical interaction between the local verifier and bank; fixed verification circuit that uses only passive linear optical components; re-usability of each note in our scheme which grows linearly with the size of note; and an unconditional security against any adversary trying to forge the banknote while tolerating the noise of up to 21.4%. We further describe a practical implementation technique of our money scheme using weak coherent states of light and the verification circuit involving a single 50/50 beam splitter and two single-photon threshold detectors. Previous best-known matching based money scheme proposal involves a verification circuit where the number of optical components increase proportional to the increase in desired noise tolerance (robustness). In contrast, we achieve any desired noise tolerance (up to a maximal threshold value) with only a fixed number of optical components. This considerable reduction of components in our scheme enables us to reach the robustness values that is not feasible for any existing money scheme with the current technology.
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.
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.