In Chapter 3, History of Integer Factorisation, Samuel S. Wagstaff, Jr gives a thorough overview of the hardness of one of the cornerstones of modern public-key cryptography. The history starts with the early effort by Eratosthenes and his sieve, eventually leading to the modern number field sieve, currently the asymptotically fastest general-purpose integer factorisation method known. Also included are 'special' integer factorisation methods like the elliptic-curve method, where the run-time depends mainly on the size of the unknown prime divisor. Modern factorisation efforts often include a gradual escalation of different methods, so it is essential to be familiar with a wide range of methods and the essence of all relevant algorithms is explained clearly.

In Chapter 5, Computing Discrete Logarithms, Robert Granger and Antoine Joux discuss the question “how hard is it to compute discrete logarithms in various groups?”. It details the key ideas and constructions behind the most efficient algorithms for solving the discrete logarithm problem (DLP) with a focus on the recent advances related to finite fields of extension degree >1. A highlight is the rapid development, in the period 2012–2014, of quasi-polynomial time algorithms to solve the DLP in finite fields of fixed charaterstic. Both Granger and Joux contributed significantly to this development, albeit on competing teams. For this book, in Chapter 5, they join forces and explain how different ideas eventually led to the fall of the fixed characteristic finite-field discrete logarithm problem.