Readings in Number Theory and Computing

Inspired by this page by @cmeik I've decided to create my own list of readings about Mathematics, specifically about Number Theory and Computing.

What I've found interesting about Number Theory is that since it mostly deals with Integer Numbers, its consequences are quite easy to "see". Besides, Number Theory is widely used for Cryptography, Random Number Generators and more, so it's quite interesting to at least have an idea of what it is about.

Books

Some of the books are purely theoretical, but most of them have direct applications in computing. I don't claim to have read all these books, but I have read several chapters or sections on each of them.

An Introduction to the Theory of Numbers

Hardy, G. H., and Edward Maitland Wright. 2008 (First ed. 1938). This is basically THE book on Number Theory. The style is very dry, it was written back in 1938 but it will guide you through almost every theorem in Number Theory. This is a mostly theoretical book.

Applied algebra for the computer sciences

Gill, Arthur. 1976. This is an old book, but quite juicy. It touches a lot of topics, like boolean algebra, sets, groups, rings and fields, error correcting codes and even graph theory. Today you can find it mostly second hand, so it's rather cheap.

A Computational Introduction to Number Theory and Algebra

Shoup, Victor. 2008. I'm really happy with this book. The explanations for the various topics are quite clear and easy to follow. You also learn about the actual applications of these theories. Also the computational part on the title already tells us that there are lot of algorithms in this book. The complexity of the running time of the algorithms is analyzed as well.

Prime numbers: a computational perspective

Crandall, Richard E., and Carl Pomerance. 2005. This is a very well presented book on the practicalities of Prime Numbers, mostly with their application to cryptography. One of the authors was the chief cryptographer at Apple, while the other author has many theorems under his name.

Groups and symmetry

Armstrong, M. A. 1988. This book presents Group Theory and Symmetry for an undergraduate level, so it's quite easy to follow and a nice intro for the other, more advanced books. Lots of pictures an examples as well.

Permutation groups

Dixon, John D., and Brian Mortimer. 1996. A more advanced introduction to Permutation Groups.

Handbook of computational group theory

Holt, Derek F., Bettina Eick, and Eamonn A. O'Brien. 2005. This book is really great. Its first chapter lays out the basics about group theory in a very nice and easy to follow manner. This is quite nice, since once you start going to the more advanced chapters you actually know what the authors are talking about. Each chapter ends with a section on the applications of what you have just read, so that's a nice thing to have.

A course in number theory and cryptography

Koblitz, Neal. 1987. This books explains most of the early cryptography and computing techniques and has an intro to elliptic curves.

The design of Rijndael: AES -- the Advanced Encryption Standard

Daemen, Joan, and Vincent Rijmen. 2002. This book is a must have. It explains the design of AES and it really shows how some areas of mathematics that look quite abstract and not so useful can be applied to design an algorithm like this one.

Concrete mathematics: a foundation for computer science

Graham, Ronald L., Donald Ervin Knuth, and Oren Patashnik. 1989. This book is a very good companion to TAOCP, giving a more detailed intro to the mathematics used in TAOCP

Hacker's Delight

Warren, Henry S. 2012. The interesting thing about this book is that it touches a lot of topics that are related to numbers and computing and how they are actually implemented in a computer, with limited RAM, limited word sizes and so on. The Number Theory of the previous books is nice, but this book shows how the actual CPU architecture plays a very important role in this kind of numeric algorithms.

The Art of Computer Programming. Volume 2, Seminumerical algorithms

Knuth, Donald Ervin. 1998. This massive volume by Knuth explains many of the algorithms related with numbers as the title shows. From how random number generators work, to working with different radixes, up to all kinds of algorithms related to polynomials.

Papers

I haven't read that many papers on the topic, since I have focused mostly on learning from books. Nonetheless here are a couple of interesting papers on Group Theory

The mathematics of perfect shuffles

Diaconis, Persi, R. L. Graham, and W. M. Kantor. 1982.

Efficient Representation of Perm Groups

Knuth, Donald E. 1991.

Online Resources

Some websites/online resources that I have used to learn about maths.

Project Euler

"Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve"

Math and Programming

by Jeremy Kun with really good examples and explanations of various topics.

Mathematics Stack Exchange

Questions and Answers about maths.

@algebrafact

Curated by @JohnDCook