*(English)*Zbl 1059.94016

The growing field of elliptic curve cryptography (ECC) has seen the publishing in the last years of several books with the common feature of gathering together mathematical materials related to the arithmetic and geometry of elliptic curves, in particular those defined over a finite ground field, together with its applications to public key cryptography. The weight of each one of these two ingredients varies in function of the objectives of each book and its foreseen reader’s typology.

The present book does not seek to put the emphasis in the mathematical aspects of the theory of elliptic curves. In the authors’ words: “the presentation is targeted to a diverse audience, and generally assumes no more than an undergraduate degree in computer science, engineering or mathematics.”

Accordingly topics such as the point counting algorithms: SEA (Schoof, Elkies and Atkins), Satoh, etc. are only mentioned (in the Preface the authors say that “presenting these topics in a readable and concise form is a formidable challenge postponed for another day”). Neither does the book approach the analysis of the specific attacks to the elliptic curve discrete logarithm problem such as the MOV (Menezes, Okamoto and Vanstone) attack, the Frey and Ruck attack or the Weil descent attack. The interested reader can find such materials in other books with a similar title, for example “Elliptic Curves in Cryptography” by *I. F. Blake, G. Seroussi* and *N. P. Smart* [Vol I (London Mathematical Society Lecture Note Series 265, Cambridge U. Press) (1999; Zbl 0937.94008); and Vol II (Cambridge U. Press) (2005)].

Instead the present book puts the emphasis in the practical side of the field to be a summary of the implementations on security aspects for ECC. Particular attention is paid to the existent industry and government ECC standards (ANSI, IEEE, and ISO/IEC) and to the algorithmic aspects of the treated topics (throughout the text, there are more than a hundred algorithms).

After an introductory chapter the book focuses on the study of efficient methods to perform the arithmetic in a finite field ${\mathbb{F}}_{q}$ (Chapter 2) and the arithmetic on an elliptic curve defined over ${\mathbb{F}}_{q}$ (Chapter 3). Chapter 4 studies some selected protocols for digital signature, public key encryption and key establishment. Finally Chapter 5 deals with some issues in software and hardware implementation as well as a paragraph that studies the side channel attacks and their countermeasures.

A paragraph of Notes at the end of each chapter provide to the reader complementary information and pertinent references (the Bibliography of the book has 489 items).