Quantum uncertainty relations are at the heart of many quantum cryptographic protocols performing classically impossible tasks. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message (the message is "locked"). Furthermore, knowing the key, it is possible to recover (or "unlock") the message.
Quantum uncertainty relations are at the heart of many quantum cryptographic protocols performing classically impossible tasks. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message (the message is "locked"). Furthermore, knowing the key, it is possible to recover (or "unlock") the message.
The existence of quantum uncertainty relations is the essential reason that some classically unrealizable cryptographic primitives become realizable when quantum communication is allowed. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking [DiVincenzo et al. 2004]. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n -bit message is encoded in a quantum system using a classical key of size much smaller than n . Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be “locked”. Furthermore, knowing the key, it is possible to recover, that is “unlock”, the message. In this article, we make the following contributions by exploiting a connection between uncertainty relations and low-distortion embeddings of Euclidean spaces into slightly larger spaces endowed with the ℓ 1 norm. We introduce the notion of a metric uncertainty relation and connect it to low-distortion embeddings of ℓ 2 into ℓ 1 . A metric uncertainty relation also implies an entropic uncertainty relation. We prove that random bases satisfy uncertainty relations with a stronger definition and better parameters than previously known. Our proof is also considerably simpler than earlier proofs. We then apply this result to show the existence of locking schemes with key size independent of the message length. Moreover, we give efficient constructions of bases satisfying metric uncertainty relations. The bases defining these metric uncertainty relations are computable by quantum circuits of almost linear size. This leads to the first explicit construction of a strong information locking scheme. These constructions are obtained by adapting an explicit norm embedding due to Indyk [2007] and an extractor construction of Guruswami et al. [2009]. We apply our metric uncertainty relations to exhibit communication protocols that perform equality testing of n -qubit states. We prove that this task can be performed by a single message protocol using O (log 2 n ) qubits and n bits of communication, where the computation of the sender is efficient.
Uncertainty relations express the fundamental incompatibility of certain observables in quantum mechanics. Far from just being puzzling constraints on our ability to know the state of a quantum system, uncertainty relations are at the heart of why some classically impossible cryptographic primitives become possible when quantum communication is allowed. This thesis is concerned with strong notions of uncertainty relations and their applications in quantum information theory.One operational manifestation of such uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be "locked". Furthermore, knowing the key, it is possible to recover, that is "unlock", the message. We give new efficient constructions of bases satisfying strong uncertainty relations leading to the first explicit construction of an informationlocking scheme. We also give several other applications of our uncertainty relations both to cryptographic and communication tasks.In addition, we define objects called QC-extractors, that can be seen as strong uncertainty relations that hold against quantum adversaries. We provide several constructions of QC-extractors, and use them to prove the security of cryptographic protocols for two-party computations based on the sole assumption that the parties' storage device is limited in transmitting quantum information. In doing so, we resolve a central question in the so-called noisy-storage model by relating security to the quantum capacity of storage devices.
Uncertainty relations give upper bounds on the accuracy by which the outcomes of two incompatible measurements can be predicted. While established uncertainty relations apply to cases where the predictions are based on purely classical data (e.g., a description of the system's state before measurement), an extended relation which remains valid in the presence of quantum information has been proposed recently [Berta et al., Nature Phys. 6, 659 (2010)]. Here, we generalize this uncertainty relation to one formulated in terms of smooth entropies. Since these entropies measure operational quantities such as extractable secret key length, our uncertainty relation is of immediate practical use. To illustrate this, we show that it directly implies security of quantum key distribution protocols. Our security claim remains valid even if the implemented measurement devices deviate arbitrarily from the theoretical model.
The thesis starts with a high-level introduction into cryptography and quantum mechanics. Chapter 2 gives a theoretical foundation by introducing probability theory, information theory, functional analysis, quantum mechanics and quantum information theory. Chapter 3, 4 and 5 are editions of work published earlier.\n\nIn Chapter 3, we present a quantum-information-theoretic tool to analyze random sampling in a quantum setting. In particular, we present two new rigorous security proofs that make use of our new sampling tool: one for BB84 quantum key distribution, and one for a quantum reduction from oblivious transfer (OT) to bit commitment.\n\nChapter 4 studies the problem of message authentication from a weak key (which is a key that is not uniformly random, e.g., a password) in a new scenario. In this scenario, the weak key is a one-time session key that is derived from a public source of randomness with the help of a long-term key (e.g., a password). We propose a new four-round protocol for message authentication from a weak (session) key. \n\nIn Chapter 5 we present a new entropic uncertainty relation and furthermore we consider the task of password-based identification. We devise a new quantum identification protocol that is secure in two security models simultaneously.
From the early days of quantum theory, it is felt that the classical ideas about information may need reorientation under a quantum theoretical framework. In quantum theory, where Heisenberg uncertainty principle is the guiding philosophy, a non-commuting observable cannot simultaneously have precisely defined values. In fact, if two observations do not commute, performing a measurement of one observable will necessarily influence the outcome of a subsequent measurement of the other observable. Hence, the act of acquiring information about a physical system inevitably disturbs the state of the system. This is because of the fact that the outcome of a measurement has some random elements due to uncertainty of the initial state of the system. Moreover, acquiring information causes a disturbance, and due to this disturbance the information cannot be copied with perfect fidelity. The information under this framework is termed as quantum information [1].
Quantum data locking is a uniquely quantum phenomenon that allows a relatively short key of constant size to (un)lock an arbitrarily long message encoded in a quantum state, in such a way that an eavesdropper who measures the state but does not know the key has essentially no information about the message. The application of quantum data locking in cryptography would allow one to overcome the limitations of the one-time pad encryption, which requires the key to have the same length as the message. However, it is known that the strength of quantum data locking is also its Achilles heel, as the leakage of a few bits of the key or the message may in principle allow the eavesdropper to unlock a disproportionate amount of information. In this paper we show that there exist quantum data locking schemes that can be made robust against information leakage by increasing the length of the key by a proportionate amount. This implies that a constant size key can still lock an arbitrarily long message as long as a fraction of it remains secret to the eavesdropper. Moreover, we greatly simplify the structure of the protocol by proving that phase modulation suffices to generate strong locking schemes, paving the way to optical experimental realizations. Also, we show that successful data locking protocols can be constructed using random code words, which very well could be helpful in discovering random codes for data locking over noisy quantum channels.