This is an excellent and timely addition to the growing body of literature on computational number theory. It has a very pleasant mix of basic theory, numerous examples that supplement the theory, and computational insights that naturally lead to the development of many clever algorithms. Best of all, the author has taken great care in collecting and weaving in historical remarks and biographical facts about the giants who laid the foundations for modern number theory as we know it today, starting from Euclid, and continuing to many contemporary contributors. This gives a good sense of the history of this rich and challenging branch of mathematics.

The book is divided into three long chapters. Chapter 1, on elementary number theory, provides a thorough introduction, and is suitable for a first introductory course. It covers most of the topics that are germane to a first course: divisibility, congruences, arithmetic functions and distribution of primes, Diophantine equations, and elliptic curves; the bread and butter of number theory. The inclusion of a section on elliptic curves is very timely, and is a welcome addition.

Chapter 2 starts with a detailed discussion of the rudiments of the computational complexity theory, and develops an overview of many of the well-known algorithms for standard problems of interest: primality testing, prime factorization, and discrete logarithms. This chapter also contains one section that provides an enjoyable introduction to quantum aspects of computing and its relevance to number theory.

The concluding chapter is devoted to one of the best-known applications of number theory: information security and cryptography. Again, this chapter provides a thorough review of applications of number-theoretic algorithms for the design of efficient arithmetic processors and public-key cryptography, digital signatures, and quantum cryptography.

This book could be used for two types of first year graduate level courses: a first course on algorithms for number theoretic problems (this could be based on chapters 1 and 2), and a second course on applications of number theory to computer arithmetic and information security. There are numerous exercises with varying degrees of difficulty. The book could also be used as a guide by professionals in the field, and is a delightful and a welcome addition to the literature.