Cryptography is concerned with the construction of schemes that should be able to withstand any abuse. Such schemes are constructed so as to maintain a desired functionality, even under malicious attempts aimed at making them deviate from their prescribed functionality.

The design of cryptographic schemes is a very difficult task. One cannot rely on intuitions regarding the typical state of the environment in which a system will operate. For sure, an adversary attacking the system will try to manipulate the environment into untypical states. Nor can one be content with countermeasures designed to withstand specific attacks, because the adversary (who will act after the design of the system has been completed) will try to attack the schemes in ways that typically will be different from the ones the designer envisioned. Although the validity of the foregoing assertions seems self-evident, still some people hope that, in practice, ignoring these tautologies will not result in actual damage. Experience shows that such hopes are rarely met; cryptographic schemes based on make-believe are broken, typically sooner rather than later.

In view of the foregoing, we believe that it makes little sense to make assumptions regarding the specific strategy that an adversary may use. The only assumptions that can be justified refer to the computational abilities of the adversary.

In this chapter we briefly discuss the goals of cryptography (Section 1.1). In particular, we discuss the basic problems of secure encryption, digital signatures, and fault-tolerant protocols. These problems lead to the notions of pseudorandom generators and zero-knowledge proofs, which are discussed as well.

Our approach to cryptography is based on computational complexity. Hence, this introductory chapter also contains a section presenting the computational models used throughout the book (Section 1.3). Likewise, this chapter contains a section presenting some elementary background from probability theory that is used extensively in the book (Section 1.2).

Finally, we motivate the rigorous approach employed throughout this book and discuss some of its aspects (Section 1.4).

Teaching Tip. Parts of Section 1.4 may be more suitable for the last lecture (i.e., as part of the concluding remarks) than for the first one (i.e., as part of the introductory remarks). This refers specifically to Sections 1.4.2 and 1.4.3.

Cryptography: Main Topics

Historically, the term “cryptography” has been associated with the problem of designing and analyzing encryption schemes (i.e., schemes that provide secret communication over insecure communication media). However, since the 1970s, problems such as constructing unforgeable digital signatures and designing fault-tolerant protocols have also been considered as falling within the domain of cryptography. In fact, cryptography can be viewed as concerned with the design of any system that needs to withstand malicious attempts to abuse it.

In this chapter we discuss pseudorandom generators. Loosely speaking, these are efficient deterministic programs that expand short, randomly selected seeds into much longer “pseudorandom” bit sequences (see illustration in Figure 3.1). Pseudorandom sequences are defined as computationally indistinguishable from truly random sequences by efficient algorithms. Hence the notion of computational indistinguishability (i.e., indistinguishability by efficient procedures) plays a pivotal role in our discussion. Furthermore, the notion of computational indistinguishability plays a key role also in subsequent chapters, in particular in the discussions of secure encryption, zero-knowledge proofs, and cryptographic protocols.

The theory of pseudorandomness is also applied to functions, resulting in the notion of pseudorandom functions, which is a useful tool for many cryptographic applications.

In addition to definitions of pseudorandom distributions, pseudorandom generators, and pseudorandom functions, this chapter contains constructions of pseudorandom generators (and pseudorandom functions) based on various types of one-way functions. In particular, very simple and efficient pseudorandom generators are constructed based on the existence of one-way permutations. We highlight the hybrid technique, which plays a central role in many of the proofs. (For the first use and further discussion of this technique, see Section 3.2.3.)

Organization. Basic discussions, definitions, and constructions of pseudorandom generators appear in Sections 3.1–3.4: We start with a motivating discussion (Section 3.1), proceed with a general definition of computational indistinguishability (Section 3.2) next present and discuss definitions of pseudorandom generators (Section 3.3), and finally present some simple constructions (Section 3.4). More general constructions are discussed in Section 3.5.

Abstract

This tribute consists of an appreciation from 1996, some further thoughts, and a list of Nash-Williams' publications.

An appreciation written on his retirement in 1996

I arrived in Aberdeen in 1965 to start my academic career as an assistant lecturer. I had become interested in graph theory and in particular in a series of papers with such resonant titles as

So it is with pleasure that I am writing this appreciation of Professor Nash-Williams in the Quincentennial Year of the University of Aberdeen.

In 1967, after some ten years at Aberdeen, Nash-Williams moved to the University of Waterloo, returning to Aberdeen as Professor of Mathematics in 1972. In 1975 he took up a Professorship of Mathematics at the University of Reading where he joined a flourishing group of combinatorialists which included Richard Rado (then recently retired), David Day kin and Anthony Hilton. He has remained at Reading ever since, apart from a year in West Virginia and frequent visits to Waterloo.

It is not my intention to give a full appreciation of Nash-Williams' contribution to graph theory. How could I? This will, I hope, be done elsewhere with a complete edition of his papers. I shall content myself with a few random remarks on his work.

Abstract

A list colouring of a graph is a colouring in which each vertex υ receives a colour from a prescribed list L(υ) of colours. This paper about list colourings can be thought of as being divided into two parts. The first part, comprising Sections 1, 2 and 6, is about proper colourings, in which adjacent vertices must receive different colours. It is a survey of known conjectures and results with few proofs, although Section 6 discusses several different methods of proof. Section 1 is intended as a first introduction to the concept of list colouring, and Section 2 discusses conjectures and results, mainly about graphs for which “ch = X”. The other part of the paper, comprising Sections 3, 4 and 5, is about improper or defective colourings, in which a vertex is allowed to have some neighbours with the same colour as itself, but not too many. Although still written mainly as a survey, this part of the paper contains a number of new proofs and new conjectures. Section 3 is about subcontractions, and includes conjectures broadly similar to Hadwiger's conjecture. Section 4 is about planar and related graphs. Section 5 is also about planar and related graphs, but this time with additional constraints imposed on the lists.

Abstract

Let Ω be a finite set and let G be a permutation group acting on Ω. The permutation group G partitions Ω into orbits. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I. The problems, given an input (Ω, G) ϵ I, are (1) count the orbits (exactly), (2) approximately count the orbits, and (3) choose an orbit uniformly at random. The goal is to quantify the computational difficulty of the problems. In particular, we would like to know for which input sets I the problems are tractable.

Introduction

Let Ω be a finite set and let G be a permutation group acting on Ω. The permutation group G partitions Ω into orbits: Two elements of Ω are in the same orbit if and only if there is a permutation in G which maps one element to the other. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I:

1. Given an input (Ω, G) ϵ I, count the orbits.

2. Given an input (Ω, G) ϵ I, approximately count the orbits.

3. Given an input (Ω, G) ϵ I, choose an orbit uniformly at random.

The goal is to quantify the computational difficulty of the problems. In particular, we would like to know for which input sets I the problems are tractable.

Many interesting orbit-counting problems come from the setting of “Polya theory”.

Abstract

“If a theorem about graphs can be expressed in terms of edges and circuits only it probably exemplifies a more general theorem about matroids.” This assertion, made by Tutte more than twenty years ago, will be the theme of this paper. In particular, a number of examples will be given of the two-way interaction between graph theory and matroid theory that enriches both subjects.

Introduction

This paper aims to be accessible to those with no previous experience of matroids; only some basic familiarity with graph theory and linear algebra will be assumed. In particular, the next section introduces matroids by showing how such objects arise from graphs. It then presents a minimal amount of theory to make the rest of the paper comprehensible. Throughout, the emphasis is on the links between graphs and matroids.

Section 3 begins by showing how 2-connectedness for graphs extends naturally to matroids. It then indicates how the number of edges in a 2-connected loopless graph can be bounded in terms of the circumference and the size of a largest bond. The main result of the section extends this graph result to matroids. The results in this section provide an excellent example of the two-way interaction between graph theory and matroid theory.

In order to increase the accessibility of this paper, the matroid technicalities have been kept to a minimum. Most of those that do arise have been separated from the rest of the paper and appear in two separate sections, 4 and 10, which deal primarily with proofs. The first of these sections outlines the proofs of the main results from Section 3.

Abstract

This paper connects the practice of wireless communication with the mathematics of quadratic forms developed by Radon and Hurwitz about a hundred years ago. Orthogonal designs, known as space-time block codes in the communications literature, provide the bridge between the two subjects. The columns of the design represent different time slots, the rows represent different transmit antennas, and the entries are the symbols to be transmitted. Multiple transmit antennas provide independent paths from the base station to the mobile terminal, and in effect this creates a single channel that is more reliable than any constituent path. The mathematics developed by Hurwitz and Radon is used to derive fundamental limits on transmission rates. The algebraic structure of the 2 × 2 space-time block code (a representation of Hamilton's biquaternions) is used to suppress interference from a second space-time user, when a second antenna is available at the mobile terminal.

Introduction

Classical coding theory is concerned with the representation of information that is to be transmitted over some noisy channel. This general framework includes the algebraic theory of error correcting codes, where codewords are strings of symbols taken from some finite field, and it includes data transmission over Gaussian channels, where codewords are vectors in Euclidean space. Fifty years of information theory and coding has led to a number of consumer products that make essential use of coding to improve reliability; for example, compact disk players, hard disk drives and wireline modems. The discovery of turbo codes by Berrou, Glavieux, and Thitmajshima [3] has led to the construction of codes that essentially achieve the Shannon capacity of the Gaussian channel.

Abstract

A survey of the most important results on partial m-systems and m-systems of finite classical polar spaces will be given. Also, the paper contains several recent results on the topic. Finally, many applications of m-systems to strongly regular graphs, linear projective two-weight codes, maximal arcs, generalized quadrangles and semi-partial geometries are mentioned.

Introduction

Let P be a finite polar space of rank r ≥ 2. An ovoid O of P is a pointset of P, which has exactly one point in common with each generator of P, that is, with each maximal totally singular subspace of P. A spread S of P is a set of generators, which constitutes a partition of the pointset. It appears that ∣O∣ = ∣S∣ for any ovoid O and any spread S of any given polar space P; this common number will be denoted by μP. Ovoids and spreads have many connections with and applications to projective planes, circle geometries, generalized polygons, strongly regular graphs, partial geometries, semi-partial geometries, codes, designs.

A partial msystem of P, with 0 ≤ m ≤ r − 1, is any set {π1, π2,…,πk} of (k≠0) totally singular m-spaces of P such that no generator containing πi has a point in common with (π1 ∪ π2 ∪ … ∪ πk) − πi,πk, with i = 1, 2,…, k. For any partial m-system M of P the bound ∣M∣ ≤ μP holds. If ∣M∣ = μP, then the partial m-system M of P is called an m-system of P.

On the occasion of the 18th British Combinatorial Conference at the University of Sussex, 1 to 6 July, 2001, this book comprises the survey papers by the nine invited speakers and a memoire of Crispin Nash-Williams, past chairman of the British Combinatorial Committee.

The survey papers range across many parts of modern combinatorics.

Martin Aigner discusses the ideas of Penrose on the 4-colour problem, as well as the application of Penrose polynomials to other combinatorial structures.

Ian Anderson surveys some of the key ideas in the study of cyclic designs, including some of the classical results of the past 150 years as well as some very recent developments.

Robert Calderbank and Ayman Naguib show the connection between the practice of wireless communication with the mathematics of quadratic forms developed by Radon and Hurwitz about a hundred years ago. This occurs through orthogonal designs, known as space-time block codes in the communications literature.

Leslie Goldberg surveys the computational problems of randomly sampling unlabelled combinatorial structures, and of counting and approximately counting unlabelled structures.

Bojan Mohar considers the interplay between graph minors and graphs embedded in surfaces.

Michael Molloy surveys the progress on two fundamental problems in random graphs and random boolean formulae. The first is the question of how many edges must be added to a random graph until it is not almost surely k-colourable.