For centuries, number theorists have refined their intuition by computing examples. The advent of computers and (especially) sophisticated algorithms has gradually led to the emergence of algorithmic number theory as a distinct field. This young discipline has been shaped by strong connections to computer science, cryptography, and other parts of mathematics. One of its charms is that mathematical ideas often lead to better algorithms. Another striking feature is that the algorithmic worldview has led to fascinating new mathematical ideas and questions.
This volume contains twenty survey articles on topics in algorithmic number theory, written by leading experts in the field. The first two are introductory, aiming to entice the reader into pursuing the subject more deeply. The next eight cover core areas of the field: factoring, primality, smooth numbers, lattices, elliptic curves, algebraic number theory, and fast arithmetic algorithms. The remaining ten articles survey specific topics, often with a distinctive perspective, including cryptography, Arakelov class groups, computational class field theory, zeta functions over finite fields, arithmetic geometry, and modular forms.