This chapter covers a more advanced topic on the mathematical foundations of cryptography: number theory. Though number theory is important to symmetric cryptography (such as in the Rijndael Cipher), it is particularly important in the public key cryptographic setting.

If you are finding the details of number theory cumbersome, I would recommend a high-level reading the first time around. You can always come back to it at a later point.

You might characterize number theory as the study of the properties of integers and mathematical functions that work with integers.

Consider, for example, that any two numbers and are coprimes (or relative primes) if their greatest common divisor equals 1. Suppose now a particular integer . How many integers smaller than are coprimes with ? Can we make general statements about the answers to this question? These are the typical types of questions that number theory seeks to answer.

Modern number theory relies on the tools of abstract algebra. The field of abstract algebra is a subdiscipline of mathematics where the main objects of analysis are abstract objects known as algebraic structures. An algebraic structure is a set of elements conjoined with one or more operations, which meets certain axioms. Through algebraic structures, mathematicians can gain insights into specific mathematical problems, by abstracting away from their details.

The field of abstract algebra is sometimes also called modern algebra. You may also come across the concept of abstract mathematics (or pure mathematics). This latter term is not a reference to abstract algebra, but rather means the study of mathematics for its own sake, and not just with an eye on potential applications.

The sets from abstract algebra can deal with many types of objects, from the shape-preserving transformations on an equilateral triangle to wallpaper patterns. For number theory, we only consider sets of elements that contain integers or functions that work with integers.