Properties of quantum mechanics such as entanglement and the no-cloning theorem enable new forms of cryptography that are unattainable in classical models of computation. Quantum information, however, is harder to process, store, and transmit than classical information. As such, it is desirable to construct semi-quantum cryptographic protocols where only a subset of the parties have quantum capabilities while the rest operate classically. In this dissertation, we analyze the security of semi-quantum protocols based on two approaches: (i) nonlocality-based techniques and (ii) the hardness of the Learning with Errors (LWE) problem. In the first part of the dissertation, we leverage a form of Bell test to construct a quantum position verification (QPV) protocol. Initial QPV protocols required verifiers to prepare and send quantum states to an honest prover to achieve security against adversaries with bounded memory or entanglement. Later work constructed QPV protocols with classical verifiers but required additional assumptions such as the hardness of LWE or of random circuit sampling. We construct a variant of QPV where classical verifiers verify the spatial locations of two provers. We use the nonlocal correlations produced by the two provers to show security against adversaries with bounded quantum memory without making any computational intractability assumptions. In the second part of the dissertation, we use trapdoor claw-free functions (TCFs) to construct semi-quantum cryptographic protocols. TCFs, which are constructible from the hardness of LWE, have been shown to be immensely valuable in the construction of classically verifiable quantum cryptographic protocols. As opposed to previous work where the server is asked to prepare a claw state and perform Pauli-X and Z measurements, we use a technique that leverages the entire range of qubit measurements on the XY-plane. We use this technique to construct three protocols: (i) a one-round blind remote state preparation protocol that delegates the preparation of an arbitrary state on the XY-plane up to a Pauli-X correction; (ii) a two-round proof of quantumness whose security is directly in terms of the hardness of LWE and only requires the prover to prepare and measure a single claw state; and (iii) a 1-of-2 Puzzle with a single qubit auxiliary state.
Properties of quantum mechanics such as entanglement and the no-cloning theorem enable new forms of cryptography that are unattainable in classical models of computation. Quantum information, however, is harder to process, store, and transmit than classical information. As such, it is desirable to construct semi-quantum cryptographic protocols where only a subset of the parties have quantum capabilities while the rest operate classically. In this dissertation, we analyze the security of semi-quantum protocols based on two approaches: (i) nonlocality-based techniques and (ii) the hardness of the Learning with Errors (LWE) problem. In the first part of the dissertation, we leverage a form of Bell test to construct a quantum position verification (QPV) protocol. Initial QPV protocols required verifiers to prepare and send quantum states to an honest prover to achieve security against adversaries with bounded memory or entanglement. Later work constructed QPV protocols with classical verifiers but required additional assumptions such as the hardness of LWE or of random circuit sampling. We construct a variant of QPV where classical verifiers verify the spatial locations of two provers. We use the nonlocal correlations produced by the two provers to show security against adversaries with bounded quantum memory without making any computational intractability assumptions. In the second part of the dissertation, we use trapdoor claw-free functions (TCFs) to construct semi-quantum cryptographic protocols. TCFs, which are constructible from the hardness of LWE, have been shown to be immensely valuable in the construction of classically verifiable quantum cryptographic protocols. As opposed to previous work where the server is asked to prepare a claw state and perform Pauli-X and Z measurements, we use a technique that leverages the entire range of qubit measurements on the XY-plane. We use this technique to construct three protocols: (i) a one-round blind remote state preparation protocol that delegates the preparation of an arbitrary state on the XY-plane up to a Pauli-X correction; (ii) a two-round proof of quantumness whose security is directly in terms of the hardness of LWE and only requires the prover to prepare and measure a single claw state; and (iii) a 1-of-2 Puzzle with a single qubit auxiliary state.
Classically testing for the presence of anti-commuting operators on a quantum device is a critical tool underpinning recent progress in classical verification of quantum computation. While such tests can be based on cryptographic assumptions, known constructions rely on highly structured assumptions, e.g. trapdoor claw-free functions.
In this work, we seek to explain this state of affairs by constructing strong cryptography from (certain forms of) classical tests of anti-commutation. In particular, we formulate the notion of a test of non-commutation (ToNC), an interactive protocol between a quantum prover and classical verifier in which the prover's final-round response is obtained by measuring one of two binary observables P0,P1 depending on the verifier's challenge bit c. We prove that, for a broad range of parameters, ToNC implies classical-communication key agreement (KA), and ToNC combined with one-way functions implies oblivious transfer (OT).
Along the way, we develop tools for and provide the first known results on hardness amplification for post-quantum KA and OT, where communication is classical but adversaries may be quantum. In particular, we prove the following results of independent interest.
- Post-quantum hard-core measure theorem: For any efficiently sampleable high-min-entropy distribution D over pairs (x,b) such that quantum circuits have advantage at most δ in predicting b from x, there exists a sub-distribution M⪯D of density (1−δ) on which b is nearly optimally quantum-hard to predict.
- Post-quantum interactive XOR lemma: Given any classically-interactive protocol, if quantum adversaries have advantage at most δ in guessing a private challenger bit b, then two sequential repetitions reduce the advantage for predicting the XOR of the challenger bits b1⊕b2 to at most δ2+negl(λ).
In the quantum computation verification problem, a quantum server wants to convince a client that the output of evaluating a quantum circuit C is some result that it claims. This problem is considered very important both theoretically and practically in quantum computation [1], [2], [3]. The client is considered to be limited in computational power, and one desirable property is that the client can be completely classical, which leads to the classical verification of quantum computation (CVQC) problem. In terms of the time complexity of server-side quantum computations (which typically dominate the total time complexity of both the client and the server), the fastest single-server CVQC protocol so far has complexity O(poly(κ)∣C∣3) where ∣C∣ is the size of the circuit to be verified and κ is the security parameter, given by Mahadev [4]. This leads to a similar cubic time blowup in many existing protocols including multiparty quantum computation, zero knowledge and obfuscation [5], [6], [7], [8], [9], [10]. Considering the preciousness of quantum computation resources, this cubic complexity barrier could be a big obstacle for theoretical and practical development of protocols for these problems.In this work, by developing new techniques, we give a new CVQC protocol with complexity O(poly(κ)∣C∣) (in terms of the total time complexity of both the client and the server), which is significantly faster than existing protocols. Our protocol is secure in the quantum random oracle model [11] assuming the existence of noisy trapdoor claw-free functions [12], which are both extensively used assumptions in quantum cryptography. Along the way, we also give a new classical channel remote state preparation protocol for states in {∣+θ⟩=21(∣0⟩+eiθπ/4∣1⟩):θ∈{0,1⋯7}}, another basic primitive in quantum cryptography. Our protocol allows for parallel verifiable preparation of L independently random states in this form (up to a constant overall error and a possibly unbounded server-side simulator), and runs in only O(poly(κ)L) time and constant rounds; for comparison, existing works (even for possibly simpler state families) all require very large or unestimated time and round complexities [13], [14], [15], [16].
This thesis deals with position-based quantum cryptography (PBQC), a type of quantum cryptography. More concretely, we focus on the design and study of quantum position verification (QPV) protocols, which act as a building block for PBQC. We prioritise the development of QPV protocols that are loss-tolerant, error-robust, and feasible with current or near-future technology, concentrating on photon-based protocols and linear-optical equipment. We also theoretically study different attack settings on QPV in the context of state discrimination, dealing with the interplay between non-locality and interaction. We introduce a new QPV protocol based on the SWAP test and Hong-Ou-Mandel interference and study it in detail regarding attacks and practical feasibility. We further provide a solution to a major practical issue of QPV -- signal loss. A minor modification of the standard structure of QPV, namely introducing a small time delay and a commitment to play from the honest prover, allows us to provably make the transmission loss between the verifiers and the honest party irrelevant for security for a class of QPV protocols. Moreover, we generalise and study QPV protocols in the continuous-variable setting. Exploring theoretical aspects of QPV attacks, we focus on the difference between quantum and classical communication. We show new bounds on existing protocols, construct protocols that can be attacked with quantum communication but not locally, show a separation between quantum and classical communication attacks on QPV, and derive a necessary condition on the attack error based on the structure of the QPV input ensemble.
Quantum mechanical effects have enabled the construction of cryptographic primitives that are impossible classically. For example, quantum copy-protection allows for a program to be encoded in a quantum state in such a way that the program can be evaluated, but not copied. Many of these cryptographic primitives are two-party protocols, where one party, Bob, has full quantum computational capabilities, and the other party, Alice, is only required to send random BB84 states to Bob. In this work, we show how such protocols can generically be converted to ones where Alice is fully classical, assuming that Bob cannot efficiently solve the LWE problem. In particular, this means that all communication between (classical) Alice and (quantum) Bob is classical, yet they can still make use of cryptographic primitives that would be impossible if both parties were classical. We apply this conversion procedure to obtain quantum cryptographic protocols with classical communication for unclonable encryption, copy-protection, computing on encrypted data, and verifiable blind delegated computation. The key technical ingredient for our result is a protocol for classically-instructed parallel remote state preparation of BB84 states. This is a multi-round protocol between (classical) Alice and (quantum polynomial-time) Bob that allows Alice to certify that Bob must have prepared n uniformly random BB84 states (up to a change of basis on his space). While previous approaches could only certify one- or two-qubit states, our protocol allows for the certification of an n-fold tensor product of BB84 states. Furthermore, Alice knows which specific BB84 states Bob has prepared, while Bob himself does not. Hence, the situation at the end of this protocol is (almost) equivalent to one where Alice sent n random BB84 states to Bob. This allows us to replace the step of preparing and sending BB84 states in existing protocols by our remote-state preparation protocol in a generic and modular way.
The promise of scalable quantum technology appears more realistic, after recent advances in both theory and experiment. Assuming a quantum computer is developed, the task of verifying the correctness of its outcome becomes crucial. Unfortunately, for a system that involves many particles, predicting its evolution via classical simulation becomes intractable. Moreover, verification of the outcome by computational methods, i.e. involving a classical witness, is believed inefficient for the hardest problems solvable by a quantum computer. A feasible alternative to verify quantum computation is via cryptographic methods, where an untrusted prover has to convince a weak verifier for the correctness of his outcome. This is the approach we take in this thesis. In the most standard configuration the prover is capable of computing all polynomial-time quantum circuits and the verifier is restricted to classical with very modest quantum power. The goal of existing verification protocols is to reduce the quantum requirements for the verifier - ideally making it purely classical - and reduce the communication complexity. In Part II we propose a composition of two existing verification protocols [Fitzsimons and Kashefi, 2012], [Aharonov et al., 2010] that achieves quadratic improvement in communication complexity, while keeping the quantum requirements for the verifier modest. Along this result, several new techniques are proposed, including the generalization of [Fitzsimons and Kashefi, 2012] to prime dimensions. In Part III we discuss the idea of model-specific quantum verification, where the prover is restricted to intermediate quantum power, i.e. between full-fledged quantum and purely classical, thus more feasible experimentally. As a proof of principle we propose a verification protocol for the One-Pure-Qubit computer [Knill and Laflamme, 1998], which tolerates noise and is capable of computing hard problems such as large matrix trace estimation. The verification protocol is an adaptation of [Fitzsimons and Kashefi, 2012] running on Measurement-Based Quantum Computing with newly proved properties of the underlying resources. Connections of quantum verification to other security primitives are considered in Part IV. Authenticated quantum communication has been already proved to relate to quantum verification. We expand this by proposing a quantum authentication protocol derived from [Fitzsimons and Kashefi, 2012] and discuss implications to verification with purely classical verifier. Connections between quantum security primitives, namely blindness - prover does not learn the computation -, and classical security are considered in Part V. We introduce a protocol where a client with restricted classical resources computes blindly a universal classical gate with the help of an untrusted server, by adding modest quantum capabilities to both client and server. This example of quantum-enhanced classical security we prove to be a task classically impossible.
Quantum computers promise to offer a considerable speed-up in solving certain problems, compared to the best classical algorithms. In many instances, the gap between quantum and classical running times is conjectured to be exponential. While this is great news for those applications where quantum computers would provide such an advantage, it also raises a significant challenge: how can classical computers verify the correctness of quantum computations? In attempting to answer this question, a number of protocols have been developed in which a classical client (referred to as verifier) can interact with one or more quantum servers (referred to as provers) in order to certify the correctness of a quantum computation performed by the server(s). These protocols are of one of two types: either there are multiple non-communicating provers, sharing entanglement, and the verifier is completely classical; or, there is a single prover and the classical verifier has a device for preparing or measuring quantum states. The latter type of protocols are, arguably, more relevant to near term quantum computers, since having multiple quantum computers that share a large amount of entanglement is, from a technological standpoint, extremely challenging. Before the realisation of practical single-prover protocols, a number of challenges need to be addressed: how robust are these protocols to noise on the verifier's device? Can the protocols be made fault-tolerant without significantly increasing the requirements of the verifier? How do we know that the verifier's device is operating correctly? Could this device be eliminated completely, thus having a protocol with a fully classical verifier and a single quantum prover? Our work attempts to provide answers to these questions. First, we consider a single-prover verification protocol developed by Fitzsimons and Kashefi and show that this protocol is indeed robust with respect to deviations on the quantum state prepared by the verifier. We show that this is true even if those deviations are the result of a correlation with the prover's system. We then use this result to give a verification protocol which is device- independent. The protocol consists of a verifier with a measurement device and a single prover. Device-independence means that the verifier need not trust the measurement device (nor the prover) which can be assumed to be fully malicious (though not communicating with the prover). A key element in realising this protocol is a robust technique of Reichardt, Unger and Vazirani for testing, using non-local correlations, that two untrusted devices share a large number of entangled states. This technique is referred to as rigidity of non-local correlations. Our second result is to prove a rigidity result for a type of quantum correlations known as steering correlations. To do this, we first show that steering correlations can be used in order to certify maximally entangled states, in a setting in which each test is independent of the previous one. We also show that the fidelity with which we characterise the state, in this specific test, is optimal. We then improve the previous result by removing the independence assumption. This then leads to our desired rigidity result. We make use of it, in a similar fashion to the device-independent case, in order to give a verification protocol that is one-sided device-independent. The importance of this application is to show how different trust assumptions affect the efficiency of the protocol. Next, we describe a protocol for fault-tolerantly verifying quantum computations, with minimal "quantum requirements" for the verifier. Specifically, the verifier only requires a device for measuring single-qubit states. Both this device, and the prover's operations are assumed to be prone to errors. We show that under standard assumptions about the error model, it is possible to achieve verification of quantum computation using fault-tolerant principles. As a proof of principle, and to better illustrate the inner workings of the protocol, we describe a toy implementation of the protocol in a quantum simulator, and present the results we obtained, when running it for a small computation. Finally, we explore the possibility of having a verification protocol, with a classical verifier and a single prover, such that the prover is blind with respect to the verifier's computation. We give evidence that this is not possible. In fact, our result is only concerned with blind quantum computation with a classical client, and uses complexity theoretic results to argue why it is improbable for such a protocol to exist. We then use these complexity theoretic techniques to show that a client, with the ability to prepare and send quantum states to a quantum server, would not be able to delegate arbitrary NP problems to that server. In other words, even a client with quantum capabilities cannot exploit those capabilities to delegate the computation of NP problems, while keeping the input, to that computation, private. This is again true, provided certain complexity theoretic conjectures are true.