In a 2009 review of the first edition of this book [1], reviewer Burkhard Englert wrote: “As an undergraduate mathematical textbook, the book is excellent from start to finish. The authors don’t just run from proof to proof; they also take the time to explain the historical background of famous techniques and algorithms.” The same can be said about the second edition: excellent from start to finish. Mathematical cryptography texts often seem to focus on interesting mathematics that just happens to be used in cryptography. In contrast, this book explains the mathematical foundations of public key cryptography in a mathematically correct and thorough way without omitting important practicalities. In addition to the usual changes linked to a new edition (correction of errors, improvements in exposition, rearrangement of topics), two new short sections were added in the final chapter on digital cash and bitcoin, and homomorphic encryption. The sections are intended to provide a quick introduction to topics that could not be developed in more detail in the main text.
The authors state in the preface: “This book provides a self-contained course for the beginning student. The only prerequisite is a first course in linear algebra.” Indeed, the book covers a lot of background material, but is perhaps not as broad and gentle as an introductory textbook on abstract algebra and number theory. Also, due to the somewhat narrow focus on public-key cryptography (Diffie-Hellman key exchange, ElGamal encryption, RSA encryption, elliptic curve cryptography, and NTRU encryption), this book may not be the perfect choice as a single text for an undergraduate cryptography course. It should at least be supplemented by a text on classical cryptography (for example, [2]).
That being said, I would like to emphasize that the book is very well written and quite clear. Topics are well motivated, and there are a good number of examples and nicely chosen exercises. To me, this book is still the first-choice introduction to public-key cryptography.
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