Randomness extraction involves the processing of purely classical information and is therefore usually studied in the framework of classical probability theory. However, such a classical treatment is generally too restrictive for applications where side information about the values taken by classical random variables may be represented by the state of a quantum system. This is particularly relevant in the context of cryptography, where an adversary may make use of quantum devices. Here, we show that the well known construction paradigm for extractors proposed by Trevisan is sound in the presence of quantum side information.We exploit the modularity of this paradigm to give several concrete extractor constructions, which, e.g., extract all the conditional (smooth) min-entropy of the source using a seed of length poly-logarithmic in the input, or only require the seed to be weakly random.

Delegating difficult computations to remote large computation facilities, with appropriate security guarantees, is a possible solution for the ever-growing needs of personal computing power. For delegated computation protocols to be usable in a larger context -or simply to securely run two protocols in parallel -the security definitions need to be composable. Here, we define composable security for delegated quantum computation. We distinguish between protocols which provide only blindness -the computation is hidden from the server -and those that are also verifiablethe client can check that it has received the correct result. We show that the composable security definition capturing both these notions can be reduced to a combination of several distinct "trace-distance-type" criteriawhich are, individually, non-composable security definitions.Additionally, we study the security of some known delegated quantum computation protocols, including Broadbent, Fitzsimons and Kashefi's Universal Blind Quantum Computation protocol. Even though these protocols were originally proposed with insufficient security criteria, they turn out to still be secure given the stronger composable definitions.

This work is intended as an introduction to cryptographic security and a motivation for the widely used Quantum Key Distribution (QKD) security definition. We review the notion of security necessary for a protocol to be usable in a larger cryptographic context, i.e., for it to remain secure when composed with other secure protocols. We then derive the corresponding security criterion for QKD. We provide several examples of QKD composed in sequence and parallel with different cryptographic schemes to illustrate how the error of a composed protocol is the sum of the errors of the individual protocols. We also discuss the operational interpretations of the distance metric used to quantify these errors.

Complex information-processing systems, for example quantum circuits, cryptographic protocols, or multi-player games, are naturally described as networks composed of more basic information-processing systems. A modular analysis of such systems requires a mathematical model of systems that is closed under composition, i.e., a network of these objects is again an object of the same type. We propose such a model and call the corresponding systems causal boxes.Causal boxes capture superpositions of causal structures, e.g., messages sent by a causal box A can be in a superposition of different orders or in a superposition of being sent to box B and box C. Furthermore, causal boxes can model systems whose behavior depends on time. By instantiating the Abstract Cryptography framework with causal boxes, we obtain the first composable security framework that can handle arbitrary quantum protocols and relativistic protocols.