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 numbergeneration 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.
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