Elliptic curves: number theory and cryptography

Abstract

present book, but contains few proofs. It should be consulted by serious students of elliptic curve cryptography. We hope that the present book provides a good introduction to and explanation of the mathematics used in that book. The books by Enge [38], Koblitz [66], [65], and Menezes [82] also treat elliptic curves from a cryptographic viewpoint and can be profitably consulted.Notation. The symbols Z, F q , Q, R, C denote the integers, the finite field with q elements, the rationals, the reals, and the complex numbers, respectively. We have used Z n (rather than Z/nZ) to denote the integers mod n. However, when p is a prime and we are working with Z p as a field, rather than as a group or ring, we use F p in order to remain consistent with the notation F q . Note that Z p does not denote the p-adic integers. This choice was made for typographic reasons since the integers mod p are used frequently, while a symbol for the p-adic integers is used only in a few examples in Chapter 13 (where we use O p ). The p-adic rationals are denoted by Q p . If K is a field, then K denotes an algebraic closure of K. If R is a ring, then R × denotes the invertible elements of R. When K is a field, K × is therefore the multiplicative group of nonzero elements of K. Throughout the book, the letters K and E are generally used to denote a field and an elliptic curve (except in Chapter 9, where K is used a few times for an elliptic integral). Acknowledgments. The author thanks Bob Stern of CRC Press forsuggesting that this book be written and for his encouragement, and the editorial staff at CRC Press for their help during the preparation of the book.