Wade Trappe - Introduction to Cryptography: With Coding Theory

with Coding Theory

3rd edition

This book is based on a course in cryptography at the upper-level undergraduate and beginning graduate level that has been given at the University of Maryland since 1997, and a course that has been taught at Rutgers University since 2003. When designing the courses, we decided on the following requirements:

• The courses should be up-to-date and cover a broad selection of topics from a mathematical point of view.

• The material should be accessible to mathematically mature students having little background in number theory and computer programming.

• There should be examples involving numbers large enough to demonstrate how the algorithms really work.

We wanted to avoid concentrating solely on RSA and discrete logarithms, which would have made the courses mostly about number theory. We also did not want to focus on protocols and how to hack into friends computers. That would have made the courses less mathematical than desired.

There are numerous topics in cryptology that can be discussed in an introductory course. We have tried to include many of them. The chapters represent, for the most part, topics that were covered during the different semesters we taught the course. There is certainly more material here than could be treated in most one-semester courses. The first thirteen chapters represent the core of the material. The choice of which of the remaining chapters are used depends on the level of the students and the objectives of the lecturer.

The chapters are numbered, thus giving them an ordering. However, except for for ease of reference; however, we recommend introducing these concepts gradually throughout the course as they are needed.

The chapters on information theory, elliptic curves, quantum cryptography, lattice methods, and error correcting codes are somewhat more mathematical than the others. The chapter on error correcting codes was included, at the suggestion of several reviewers, because courses that include introductions to both cryptology and coding theory are fairly common.

Suppose you want to give an example for RSA. You could choose two one-digit primes and pretend to be working with fifty-digit primes, or you could use your favorite software package to do an actual example with large primes. Or perhaps you are working with shift ciphers and are trying to decrypt a message by trying all 26 shifts of the ciphertext. This should also be done on a computer.

Additionally, at the end of the book are appendices containing computer examples written in each of Mathematica, Maple, MATLAB, and Sage that show how to do such calculations. These languages were chosen because they are user friendly and do not require prior programming experience. Although the course has been taught successfully without computers, these examples are an integral part of the book and should be studied, if at all possible. Not only do they contain numerical examples of how to do certain computations but also they demonstrate important ideas and issues that arise. They were placed at the end of the book because of the logistic and aesthetic problems of including extensive computer examples in these languages at the ends of chapters.

Additionally, programs available in Mathematica, Maple, and MATLAB can be downloaded from the Web site (bit.ly/2JbcS6p). Homework problems (the computer problems in various chapters) based on the software allow students to play with examples individually. Of course, students having more programming background could write their own programs instead. In a classroom, all that is needed is a computer (with one of the languages installed) and a projector in order to produce meaningful examples as the lecture is being given.