In this thesis we explore communication between parties with access to quantum resources, such as channels, qudits, and computers. We start by studying types of quantum channels. In particular, we consider a scenario where the sender knows a classical description of the qudit they intend to send, and the receiver’s operations are restricted to classical ones. Our main result is that the accuracy of the transmission scales inverse exponentially with the number of pre-shared entangled qudits. We later look into possible extra properties of quantum channels by giving a protocol for authenticating a noisy channel. Moreover, we prove that our protocol requires access to poly-logarithmic fewer qubits than the previously known techniques. For the rest of the dissertation we look at what ideal quantum channels could be useful for. Our first result is a round-optimal quantum protocol for oblivious transfer which can be instantiated both in the plain and quantum random oracle models (by basically lifting the properties of an underlying zero-knowledge protocol), but we obtain round optimality in the quantum random oracle model only. The last chapters of this dissertation are an exploration of a particular set-up assumption, quantum pseudorandomness; i.e. assuming the existence of random (looking) quantum states. We first show that in contrast to the classical case, the size of a quantum pseudorandom object cannot be shrunk. Finally, we prove that if there is a promise problem that admits a quantum reduction that loses information about its input, then certain quantum pseudorandom primitives exit.
In this thesis we explore communication between parties with access to quantum resources, such as channels, qudits, and computers. We start by studying types of quantum channels. In particular, we consider a scenario where the sender knows a classical description of the qudit they intend to send, and the receiver’s operations are restricted to classical ones. Our main result is that the accuracy of the transmission scales inverse exponentially with the number of pre-shared entangled qudits. We later look into possible extra properties of quantum channels by giving a protocol for authenticating a noisy channel. Moreover, we prove that our protocol requires access to poly-logarithmic fewer qubits than the previously known techniques. For the rest of the dissertation we look at what ideal quantum channels could be useful for. Our first result is a round-optimal quantum protocol for oblivious transfer which can be instantiated both in the plain and quantum random oracle models (by basically lifting the properties of an underlying zero-knowledge protocol), but we obtain round optimality in the quantum random oracle model only. The last chapters of this dissertation are an exploration of a particular set-up assumption, quantum pseudorandomness; i.e. assuming the existence of random (looking) quantum states. We first show that in contrast to the classical case, the size of a quantum pseudorandom object cannot be shrunk. Finally, we prove that if there is a promise problem that admits a quantum reduction that loses information about its input, then certain quantum pseudorandom primitives exit.
Quantum computers can make use of quantum mechanics to achieve surprising speed-ups relative to classical computers, for important computational problems. Quantum computers are gradually coming into reality, but the current implementations are big, require advanced facilities and could only be used as a cloud service in the foreseeable future. If we want to run quantum computations on secret data, we require protocols for performing computations on a remote quantum server without leaking information to the server about the data being processed. In the quantum computation delegation problem, a client with limited quantum computing resources wishes to evaluate a secret quantum circuit C with the help of a more powerful remote server. The protocol is deemed secure if the server learns nothing about either the circuit or the input state even if the server acts maliciously. This problem, first raised in (Childs05), has become fundamental to the study of quantum cryptography, because of its practical importance, and because it provides a testbed for new techniques that can be later applied to related problems (for example, quantum computation verification [FK12]). Prior to our work, known protocols for this problem were either proven secure against all possible attackers, without any complexity-theoretic assumptions, but highly inefficient for the client; or efficient but proven secure only based on strong cryptographic assumptions. This second category of protocols is only secure if there exist public-key encryption schemes that are secure against quantum attacks and that have special additional properties. Very few candidates for such schemes are known, and their security remains poorly understood. We seek to understand how efficient quantum delegation schemes can be designed based on weaker or different complexity assumptions. We study how the availability of symmetric-key primitives, modeled by a random oracle, changes the complexity of quantum computation delegation. In contrast with public-key primitives, there are many candidates for symmetric-key schemes secure against quantum attacks. We aim for protocols that require as little quantum computing as possible for the client. Specifically, we ask that the client only needs to run a quantum circuit of size independent of the circuit being evaluated. Known unconditionally-secure protocols all require the client to evaluate quantum circuits of size at least linear in the original circuit size. In this thesis, we construct two new quantum computation delegation protocols. The first protocol is for the delegation of a large special circuit family, which we call "C+P circuits". This scheme is non-interactive, and can be used to delegate a very important quantum algorithm---Shor's algorithm (Shor,97) for integer factorization and discrete logarithms. Our second protocol requires interaction---one quantum message from the client to the server, and many rounds of classical computation---but supports arbitrary quantum computation (instead of a specific circuit family), and is thus much more general than the first protocol in both theory and practice. Both protocols are proved secure in the quantum random oracle model. The protocols can be instantiated in practice with a symmetric-key cipher such as AES.
Quantum information theory studies the fundamental limits that physical laws impose on information processing tasks such as data compression and data transmission on noisy channels. This thesis presents general techniques that allow one to solve many fundamental problems of quantum information theory in a unified framework. The central theorem of this thesis proves the existence of a protocol that transmits quantum data that is partially known to the receiver through a single use of an arbitrary noisy quantum channel. In addition to the intrinsic interest of this problem, this theorem has as immediate corollaries several central theorems of quantum information theory. The following chapters use this theorem to prove the existence of new protocols for two other types of quantum channels, namely quantum broadcast channels and quantum channels with side information at the transmitter. These protocols also involve sending quantum information partially known by the receiver with a single use of the channel, and have as corollaries entanglement-assisted and unassisted asymptotic coding theorems. The entanglement-assisted asymptotic versions can, in both cases, be considered as quantum versions of the best coding theorems known for the classical versions of these problems. The last chapter deals with a purely quantum phenomenon called locking. We demonstrate that it is possible to encode a classical message into a quantum state such that, by removing a subsystem of logarithmic size with respect to its total size, no measurement can have significant correlations with the message. The message is therefore "locked" by a logarithmic-size key. This thesis presents the first locking protocol for which the success criterion is that the trace distance between the joint distribution of the message and the measurement result and the product of their marginals be sufficiently small.
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.
We consider the problem of implementing two-party interactive quantum communication over noisy channels, a necessary endeavor if we wish to fully reap quantum advantages for communication. For an arbitrary protocol with n messages, designed for noiseless qudit channels (where d is arbitrary), our main result is a simulation method that fails with probability less than 2−Θ (nє) and uses a qudit channel n (1 + Θ (√є)) times, of which an є fraction can be corrupted adversarially. The simulation is thus capacity achieving to leading order, and we conjecture that it is optimal up to a constant factor in the √є term. Furthermore, the simulation is in a model that does not require pre-shared resources such as randomness or entanglement between the communicating parties. Perhaps surprisingly, this outperforms the best known overhead of 1 + O(√є loglog1/є) in the corresponding classical model, which is also conjectured to be optimal [Haeupler, FOCS’14]. Our work also improves over the best previously known quantum result where the overhead is a non-explicit large constant [Brassard et al., FOCS’14] for low є.
There is a large body of work studying what forms of computational hardness are needed to realize classical cryptography. In particular, one-way functions and pseudorandom generators can be built from each other, and thus require equivalent computational assumptions to be realized. Furthermore, the existence of either of these primitives implies that P ≠ N P , which gives a lower bound on the necessary hardness.One can also define versions of each of these primitives with quantum output: respectively one-way state generators and pseudorandom state generators. Unlike in the classical setting, it is not known whether either primitive can be built from the other. Although it has been shown that pseudorandom state generators for certain parameter regimes can be used to build one-way state generators, the implication has not been previously known in full generality. Furthermore, to the best of our knowledge, the existence of one-way state generators has no known implications in complexity theory.We show that pseudorandom states compressing n bits to log n + 1 qubits can be used to build one-way state generators and pseudorandom states compressing n bits to ω ( log n ) qubits are one-way state generators. This is a nearly optimal result since pseudorandom states with fewer than c log n -qubit output can be shown to exist unconditionally. We also show that any one-way state generator can be broken by a quantum algorithm with classical access to a P P oracle.An interesting implication of our results is that a t ( n ) -copy one-way state generator exists unconditionally, for every t ( n ) = o ( n / log n ) . This contrasts nicely with the previously known fact that O ( n ) -copy one-way state generators require computational hardness. We also outline a new route towards a black-box separation between one-way state generators and quantum bit commitments.
How could quantum cryptography help us achieve what are not achievable in classical cryptography? In this work we study the classical cryptographic problem that two parties would like to perform secure computations with long outputs. As a basic primitive and example, we first consider the following problem which we call secure function sampling with long outputs: suppose f:{0,1}n→{0,1}m is a public, efficient classical function, where m is big; Alice would like to sample x from its domain and sends f(x) to Bob; what Bob knows should be no more than f(x) even if it behaves maliciously. Classical cryptography, like FHE and succinct arguments [Gen09,Kil92,HW15], allows us to achieve this task within communication complexity O(n+m); could we achieve this task with communication complexity independent of m? In this work, we first design a quantum cryptographic protocol that achieves secure function sampling with approximate security, within O(n) communication (omitting the dependency on the security parameter and error tolerance). We also prove the classical impossibility using techniques in [HW15], which means that our protocol indeed achieves a type of quantum advantage. Building on the secure function sampling protocol, we further construct protocols for general secure two-party computations [Yao86,GB01] with approximate security, with communication complexity only depending on the input length and the targeted security. In terms of the assumptions, we construct protocols for these problems assuming only the existence of collapsing hash functions [Unr16]; what's more, we also construct a classical-channel protocol for these problems additionally assuming the existence of noisy trapdoor claw-free functions [BCMVV,BKVV].