Page 13 - Hardware Implementation of Finite-Field Arithmetic

xii Pr ef a c e

• Most algorithms are described in Ada, a programming
language similar to VHDL, so that they can be executed and
the correctness of the proposed algorithms can be verified
with actual input data.
• In what concerns the description of the circuits, logic schemes
are presented as well as VHDL models, in such a way that
the corresponding circuits can be easily simulated and
synthesized.

Overview
The book is divided into 10 chapters. The first chapter (mathematical
background) gives the main definitions and properties of finite fields.
Chapters 2 to 4 are dedicated to the operations modulo m and the
corresponding circuits. Chapter 2 deals with the modulo m reduction,
Chap. 3 with the modulo m addition, subtraction, multiplication, and
exponentiation, and Chap. 4 with the modulo p division, where p is a
prime. Chapters 5 and 6 are dedicated to the operations modulo f(x),
where f(x) is a polynomial over a finite field, and to the corresponding
circuits. Chapter 5 deals with the modulo f(x) addition, subtraction,
multiplication, and exponentiation, and Chap. 6 with the modulo f(x)
division, where f(x) is an irreducible polynomial. Chapters 7 to 9 are
m
dedicated to the main arithmetic operations over GF(2 ). In Chap. 7
polynomial bases are considered (thus, a particular case of the topics
dealt with in Chaps. 5 and 6). In Chap. 8 normal bases are used, and
in Chap. 9 dual and triangular bases are considered. Chapter 10 is
dedicated to elliptic-curve cryptography, currently one of the main
finite-field applications.
There are four appendices. Three of them describe circuits for
performing arithmetic operations over some particular fields, namely
a prime field GF(2 − 2 − 1) in App. A, two optimal extension fields
64
192
GF(239 ) and GF((2 − 387) ) in App. B, and two binary extension
6
32
17
163
233
fields GF(2 ) and GF(2 ) in App. C. Appendix D is a brief comparison
of the syntaxes of Ada and VHDL.
All the chapters, but the first one, include algorithms, circuits,
and results of FPGA implementations. The algorithms are described
in Ada and the circuits are modeled in VHDL. Complete and
executable source files (Ada and VHDL) are available at the authors’
Web site www.arithmetic-circuits.org.