In this chapter we consider the problem of amplifying secrecy, or uncertainty. This is the problem of privacy amplification: given a partially secret string, how do we make it almost-perfectly secret? This task forms one of the key building blocks in the protocol for quantum key distribution that we develop in later chapters. It can be solved using a fundamental object from the theory of pseudorandomness called a randomness extractor. We introduce an extractor based on hashing and show that it can be used to perform privacy amplification.

Quantum information provides an advantage for cryptographic tasks in a wide variety of settings. In this chapter we focus on applications involving two parties and look beyond key distribution for other tasks where quantum information can play a role. This includes coin flipping, oblivious transfer, and other primitives in two-party cryptography.

Chapter 2 summarised the many and varied factors that can contribute to cancer. A critical point to emerge is that the causes fall into two broad classes: those over which we have control and those we have to live with. Prominent among the former are tobacco use, excessive consumption of red and processed meats, and pollution, particularly of drinking water. It’s a striking fact that elimination of these would reduce the global cancer burden by at least three-quarters. The fact that, although cancers have multiple causes, they all appear to be driven by mutations in DNA focuses attention on the sequence of bases in DNA, and the acquisition of massive amounts of sequence data for tumours has become a cornerstone of current cancer research.

In this book, we have seen how the cancer pathway has threaded its way through human history for four millennia, its course being marked by milestones of major advances that have saved many lives and offered much hope. It is also littered with the skeletons of failed experiments and dashed optimism. In painting this contemporary picture the chapters have summarised: (1) global incidence and mortality, (2) causes, (3) DNA sequencing, (4) signalling pathways in normal cells, (5) subversion of signalling by mutations, (6) what makes a tumour, (7) major signalling pathways in tumour cells and (8) cancer detection.

Chapter 8 reviewed the range of methods in use for tumour detection and radiation therapy together with the continuing search for biomarkers to detect the presence of tumours and monitor their progression in response to therapies. In this chapter, we reach what has in many ways become the central goal of cancer research, namely chemotherapy – the notion that it is possible to bring cancers under control through the use of drugs without the requirement for surgical intervention or radiotherapy. The science of treating cancer with specific chemicals was initiated in the second half of the twentieth century as an adjunct to surgery and radiotherapy. Starting from the use of single agents and advancing to the administration of drug cocktails, they gave spectacular results for some types of cancer. By the beginning of the next millennium, these anti-cancer drugs had been joined by the first ‘targeted’ agents, namely kinase inhibitors and the first monoclonal antibodies. The astonishing success of vaccines that prevent infection by human papillomaviruses (HPVs) has had a major impact on the incidence of cervical carcinoma and other types of cancer caused by oncogenic forms of these viruses.

Epidemiology is the study of the distribution and determinants (causes and risk factors) of health-related states and events. This chapter focuses on the distribution of cancers across the world; the next chapter deals with causes. We begin by looking at global patterns. These are of interest because they show marked variations in the forms of the disease that afflict different populations. These differences indicate the importance of environmental factors that include lifestyle – for example, what we eat and tobacco use – in determining both the type of cancer and the frequency of occurrence.

“We revisit the quantum one-time pad and investigate the possibility of shortening the key used for quantum encryption. We first provide an impossibility result, and then show how it can be circumvented in two different ways: using approximate encryption, and by opening the door to the fascinating world of computational security. We also discuss a new possibility for quantum encryption, which is known as certified deletion: this is the possibility for the encrypter of a secret to request that the ciphertext is provably and irrevocably erased.”

Chapter 1 surveyed the global picture of cancer incidence and mortality, revealing that across nations there is an almost sixfold range in new cases per year. The mortality variation is somewhat smaller at just over threefold, but, taken together, these figures indicate that multiple factors give rise to cancers and that there are considerable differences in the efficacy of treatment. The current figures of 18 million diagnoses and 10 million deaths each year are predicted to increase by over 60% in the next 20 years. The human implications of these figures are beyond comprehension, but the economic costs are also staggering. In the USA alone, cancer research funding runs to billions of dollars a year and the annual cost of caring for cancer patients to hundreds of billions.

Entanglement is one of the most fundamental, and intriguing, properties of quantum mechanics. It is also at the heart of quantum cryptography! In this chapter we start by giving a clear mathematical definition of entanglement. We give two classic applications, to superdense coding and to secret sharing. We then investigate two complementary properties of entanglement that we will use deeply in cryptographic applications. The first is nonlocality, which we investigate through the famous CHSH game. The second is the monogamy of entanglement, which we demonstrate using a three-player version of the CHSH game.