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Introduction to Cryptography with Mathematical Foundations and Computer Implementations

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This book is in the MAA's basic library list.

MAA Review

There are many introductory cryptography textbooks on the market — as of this writing, I count at least 15 such books in the MAA Reviews database. Different books choose to emphasize different aspects of the material: some books focus on the mathematical, some on the historical, and some on the computer science aspects of the field. As such, each book has slightly different things to offer the reader, and which one is the most appropriate will depend greatly on a given reader’s interests and background. One of the more recent entries into the field is Alexander Stanoyevitch’s Introduction to Cryptography, and I found it to be a particularly good entry in a crowded field.

Stanoyevitch’s book was designed as a textbook for sophomore-level math and computer science majors, so it has very little in the way of technical prerequisites (he defines modular arithmetic and how to multiply matrices, for example) but does assume a certain level of mathematical maturity and motivation that many introductory texts do not. The topics covered in the book include all of the cryptosystems that one might expect: Vigenere, Playfair, One Time Pads, The Enigma Machine, Hill Ciphers, DES, RSA, AES, Knapsack ciphers, and Elliptic Curve Cryptosystems. Of course, covering all of these cryptosystems involves introducing quite a few ideas from Number Theory, Linear Algebra, Combinatorics, and Abstract Algebra, and Stanoyevitch does a good job of introducing just enough of those topics to cover the cryptosystems without delving into more detail than necessary. Along the way, he also details much of the history of the topics, and characters like Babbage, Rejewski, Turing, Diffie, and Hellman all make appearances. As someone who has taught cryptography courses in the past, I was particularly impressed with the scaled-down versions of DES and AES that the author describes, which seem to do a good job of capturing the key ideas behind these cryptosystems while still keeping them at a small enough size that one could actually demonstrate how they work.

Stanoyevitch’s writing style is clear and engaging, and the book has many examples illustrating the mathematical concepts throughout. Every chapter comes with a large number of exercises ranging from the computational (“Find the prime factorization of 74529”, “Use the RSA algorithm with public key (69353,4321) to encrypt the plaintext message 12345”) to the theoretical (“Show that –1 has a square root modulo a prime p if and only if p ≡ 1 (mod 4)”, “Verify that any three-round Feistel cryptosystem is self-decrypting”). Many of these exercises have worked out solutions in the rear of the book. One of the many smart decisions that the author made was to also include many computer implementations and exercises at the end of each chapter. These are separated from the main text and could easily be skipped by readers looking for a mostly theoretical point of view on the material, but could also give readers with more programming background some good opportunities to get heir hands dirty (so to speak). It is also worth noting that he has many Matlab implementations of his own on his website. All of these exercises and suggested computer implementations give the reader an opportunity to further their understanding of some material and introduce them to a large number of other topics.

It is clear that Stanoyevitch designed this book to be used by students and that he has taught this type of student many times before. The book feels carefully structured in a way that builds nicely, interspersing the less-technical material with the more-technical in a way that students will appreciate, and covering all the background material to give the reader answers to their questions just before they need them. Stanoyevitch’s book will not be the best choice for everyone who wants to learn cryptography, but it is definitely a solid choice and will be on the short list of books that I would recommend to a student wanting to learn about the field.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College, where he has been teaching a first year seminar on cryptography since 2007. He can be reached at dglass@gettysburg.edu.

BLL — The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

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