Introduction

Abstract

To be involved with modern cryptography is to dive willy-nilly into number theory, that is, the study of the natural numbers, one of the most beautiful areas of mathematics. However, we have no intention of becoming deep-sea divers who raise sunken treasure from the mathematical ocean floor, which in any case is unnecessary for cryptographic applications. Our goals are much more modest. On the other hand, there is no limit to the depth of involvement of number theory with cryptography, and many significant mathematicians have made important contributions to this area.

God created the integers. All the rest is the work of man.

—Leopold Kronecker

If you look at zero you see nothing; but look through it and you will see the world.

—Robert Kaplan, The Nothing That Is: A Natural History of Zero