The discrete logarithm problem based on elliptic and hyperelliptic curves has gained a lot of popularity as a cryptographic primitive. The main reason is that no subexponential algorithm for computing discrete logarithms on small genus curves is currently available, except in very special cases. Therefore curve-based cryptosystems require much smaller key sizes than RSA to attain the same security level. This makes them particularly attractive for implementations on memory-restricted devices like smart cards and in high-security applications.
The
Handbook of Elliptic and Hyperelliptic Curve Cryptography introduces the theory and algorithms involved in curve-based cryptography. After a very detailed exposition of the mathematical background, it provides ready-to-implement algorithms for the group operations and computation of pairings. It explores methods for point counting and constructing curves with the complex multiplication method and provides the algorithms in an explicit manner. It also surveys generic methods to compute discrete logarithms and details index calculus methods for hyperelliptic curves. For some special curves the discrete logarithm problem can be transferred to an easier one; the consequences are explained and suggestions for good choices are given. The authors present applications to protocols for discrete-logarithm-based systems (including bilinear structures) and explain the use of elliptic and hyperelliptic curves in factorization and primality proving. Two chapters explore their design and efficient implementations in smart cards. Practical and theoretical aspects of side-channel attacks and countermeasures and a chapter devoted to (pseudo-)random number generation round off the exposition.
The broad coverage of all- important areas makes this book a complete handbook of elliptic and hyperelliptic curve cryptography and an invaluable reference to anyone interested in this exciting field.
Customer Reviews & Comments
Elliptic curve cryptography is now an entrenched field and has been subjected to an enormous amount of research in the last fifteen years. As soon as encryption schemes based on arithmetic in elliptic curves were proposed, it was natural to speculate on whether these schemes could be generalized to hyperelliptic curves or even general abelian varieties. This book gives an overview of what has been done, and even though most of the proofs are omitted, it does serve a need for those interested in the latest developments in the subject. This reviewer did not read the entire book, but concentrated instead on only a few parts that discussed developments in the last few years. Just skimming the book though will reveal that the authors have been very thorough in giving the reader the necessary mathematical background for a study of ECC and HECC cryptography. Readers needing more detailed background can consult the many references.
