Elliptic Curve Cryptography
and Scott Vanstone
The study of elliptic curves by algebraists, algebraic geometers and
number theorists dates back to the middle of the nineteenth century,
and there now exists an extensive literature that describes the
beautiful and elegant properties of these marvelous objects.
In 1984, Hendrik Lenstra described an ingenious algorithm for factoring
integers that relies on properties of elliptic curves. This discovery
prompted researchers to investigate other applications of elliptic
curves in cryptography and computational number theory.
Public-key cryptography was conceived in 1976 by Whitfield Diffie and
Martin Hellman. The first practical realization followed in 1977 when
Ron Rivest, Adi Shamir and Len Adleman proposed their now well-known
RSA cryptosystem, whose security is based on the intractability of
the integer factorization problem. Elliptic curve cryptography (ECC)
was discovered in 1985 by Neal Koblitz and Victor Miller.
Elliptic curve cryptographic schemes are public-key mechanisms
that provide the same functionality as RSA schemes. However, their
security is based on the hardness of a different problem, namely
the elliptic curve discrete logarithm problem (ECDLP). Currently the best
algorithms known to solve the ECDLP have fully exponential running
time, in contrast to the subexponential-time algorithms known for
the integer factorization problem. This means that a desired security
level can be attained with significantly smaller keys in
elliptic curve systems than is possible with their RSA counterparts.
For example, it is generally accepted that a 160-bit elliptic
curve key provides the same level of security as a 1024-bit
RSA key. The advantages that can be gained from smaller key sizes
include speed and efficient use of power, bandwidth, and storage.
This book is intended as a guide for security professionals,
developers, and those interested in learning how elliptic
curve cryptography can be deployed to secure applications.
Most of the material should be accessible to anyone with
an undergraduate degree in computer science, engineering,
or mathematics. The book was not written for theoreticians as
is evident from the lack of proofs for mathematical statements.
However, the breadth of coverage and the extensive surveys of
the literature included at the end of each chapter should make it
a useful resource for the researcher.
The book has a strong focus on efficient methods for finite field
arithmetic (Chapter 2) and elliptic curve arithmetic
(Chapter 3). Chapter 4 surveys the known attacks
on the ECDLP, and describes the generation and validation of domain
parameters and key pairs, and selected elliptic curve protocols
for digital signature, public-key encryption and key establishment.
We chose not to include the mathematical details of the attacks
on the ECDLP, or descriptions of algorithms for counting the
points on an elliptic curve, because the relevant mathematics
is quite sophisticated. (Presenting these topics in an accessible
and concise form is a formidable challenge left for another day.)
The choice of material in Chapters 2, 3
and 4 was heavily influenced by the contents of ECC standards
that have been developed by accredited standards bodies, in particular
the FIPS 186-2 standard for the Elliptic Curve Digital Signature
Algorithm (ECDSA) developed by the U.S. government's National
Institute for Standards and Technology (NIST). Chapter 5 details
selected aspects of efficient implementations in software and hardware,
and also gives an introduction to side-channel attacks and their
countermeasures. Although the coverage in Chapter 5 is admittedly
narrow, we hope that the treatment provides a glimpse of engineering
considerations faced by software developers and hardware designers.
We gratefully acknowledge the following people who provided
valuable comments and advice:
Edlyn Teske, and
A special thanks goes to Helen D'Souza,
whose artwork graces several pages of this book.
Thanks also to Cindy Hankerson and Sherry Shannon-Vanstone for
suggestions on the general theme of "curves in nature" represented
in the illustrations. Finally, we would like to thank our editors at
Springer, Wayne Wheeler and Wayne Yuhasz, for their continued
encouragement and support.
Updates, errata, and our contact information are available at our
greatly appreciate if readers informed us of the inevitable errors
Darrel R. Hankerson, Alfred J. Menezes, Scott A. Vanstone
Auburn & Waterloo
Guide to Elliptic Curve Cryptography /
Illustration by Helen D'Souza