Page 12 - Hardware Implementation of Finite-Field Arithmetic

Preface

inite fields are used in different types of computers and digital
communication systems. Two well-known examples are error-
Fcorrection codes and cryptography. The traditional way of imple-
menting the corresponding algorithms is software, running on
general-purpose processors or on digital-signal processors. Never-
theless, in some cases the time constraints cannot be met with instruction-
set processors, and specific hardware must be considered, that is,
circuits specifically designed for executing those complex algorithms:
they implement the particular computation primitives of the algo-
rithms and profit from their inherent parallelism.
Apart from the application-specific integrated circuits (ASICs)
solution, another technology at hand for developing specific circuits is
constituted by field-programmable gate arrays (FPGA). They form an
attractive option for small production quantities as their nonrecurrent
engineering costs are much lower than those corresponding to ASICs.
They also offer flexibility and fast time-to-market. Furthermore, in
order to reduce their size, and so the unit cost, an interesting possibility
is to reconfigure them at run time so that the same programmable
device can execute different predefined functions.
This book describes algorithms and circuits for executing the main
finite-field operations, that is, addition, subtraction, multiplication,
squaring, exponentiation, and division. It is mainly addressed to hard-
ware engineers involved in the development of embedded systems,
including finite-field operations. Distinguishing features of this book
are the following:
• The emphasis is different from the classic texts on finite fields.
It is not limited to the description of algebraic and algorithmic
aspects. The main topic is circuit synthesis.
• A special importance has been given to FPGA implementa-
tions. The particular architecture of these components leads
the designer to use synthesis techniques somewhat different
than the ones applied for ASIC for which standard cell libraries
exist. Throughout the book examples of FPGA implementa-
tion are described.
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