Das A., Madhavan C.E.V. Public-key Cryptography: Theory and Practice

Pearson, 2009. — 585 p.Most cryptography textbooks today, even many of the celebrated ones, essentially take a narrative approach. While such an approach may be suitable for beginners at an undergraduate level, it misses the finer details in this rapidly growing area of applied mathematics. The fact that public-key cryptography is mathematical is hard to deny and a mathematical subject would be better treated in the mathematical way.
This is precisely the point that this book addresses, that is, it proceeds in a canonically mathematical way while revealing cryptographic concepts. This mathematics is often not so simple (and that is why other textbooks didn’t bother to mention it), but we plan to stick to mathematical sophistication as far as possible. A typical feature of this book is that it does not rely on anything other than the readers’ mathematical intuitions; it develops all the mathematical abstractions starting from scratch. Although computer science and mathematics students nowadays do undergo some courses on discrete structures somewhere in their curricula, we do not assume this; instead we develop the algebra starting at the level of set operations. Simpler structures like groups, rings and fields are followed by more complex concepts like finite fields, algebraic curves, number fields and p-adic numbers. The resulting (long) compilation of abstract mathematical tools tends to relieve cryptography students and researchers from consulting many mathematics books for understanding the background concepts. We are happy to offer this self-sufficient treatment complete with proofs and other details. The only place where we had to be somewhat sketchy is the discussion on elliptic and hyperelliptic curves. The mathematics here seems to be too vast to fit in a few pages and we opted for a deliberate simplification of these topics.