Introduction to Cryptography with Coding Theory, 3rd Edition

3.11 Finite Fields

Note: This section is more advanced than the rest of the chapter. It is included because finite fields are often used in cryptography. In particular, finite fields appear in four places in this book. The finite field $GF(2^8)$ is used in AES (Chapter 8). Finite fields give an explanation of some phenomena that are mentioned in Section 5.2. Finally, finite fields are used in Section 21.4, Chapter 22 and in error correcting codes (Chapter 24).

Many times throughout this book, we work with the integers mod $p\text{,}$ where $p$ is a prime. We can add, subtract, and multiply, but what distinguishes working mod $p$ from working mod an arbitrary integer $n$ is that we can divide by any number that is nonzero mod $p\text{.}$ For example, if we need to solve ...