and Overview.- Finite Field Arithmetic.- Elliptic Curve Arithmetic.- Cryptographic Protocols.- Implementation Issues.
Roadmap (p. 19-20)
Before implementing an elliptic curve system, several selections have to be made concerning the finite field, elliptic curve, and cryptographic protocol:
1. a finite field, a representation for the field elements, and algorithms for performing field arithmetic;
2. an elliptic curve, a representation for the elliptic curve points, and algorithms for performing elliptic curve arithmetic; and
3. a protocol, and algorithms for performing protocol arithmetic.
There are many factors that can infiuence the choices made. All of these must be considered simultaneously in order to arrive at the best solution for a particular application. Relevant factors include security considerations, application platform (software or hardware), constraints of the particular computing environment (e.g., processing speed, code size (ROM), memory size (RAM), gate count, power consumption), and constraints of the particular communications environment (e.g., bandwidth, response time).
Not surprisingly, it is difficult, if not impossible, to decide on a single "best" set of choices. For example, the optimal choices for a workstation application can be quite different from the optimal choices for a smart card application. The purpose of this book is to provide security practitioners with a comprehensive account of the various implementation and security considerations for elliptic curve cryptography, so that informed decisions of the most suitable options can be made for particular applications. The remainder of the book is organized as follows.
Chapter 2 gives a brief introduction to finite fields. It then presents algorithms that are well-suited for software implementation of the arithmetic operations in three kinds of finite fields-prime fields, binary fields and optimal extension fields.
Chapter 3 provides a brief introduction to elliptic curves, and presents different methods for representing points and for performing elliptic curve arithmetic. Also considered are techniques for accelerating the arithmetic on Koblitz curves and other elliptic curves admitting efficiently-computable endomorphisms.
Chapter 4 describes elliptic curve protocols for digital signatures, public-key encryption and key establishment, and considers the generation and validation of domain parameters and key pairs. The state-of-the-art in algorithms for solving the elliptic curve discrete logarithm problem are surveyed.
Chapter 5 considers selected engineering aspects of implementing elliptic curve cryptography in software and hardware. Also examined are side-channel attacks where an adversary exploits information leaked by cryptographic devices, including electromagnetic radiation, power consumption, and error messages.
The appendices present some information that may be useful to implementors. Appendix A presents specific examples of elliptic curve domain parameters that are suitable for cryptographic use. Appendix B summarizes the important standards that describe elliptic curve mechanisms. Appendix C lists selected software tools that are available for performing relevant number-theoretic calculations.
