Efficient Algorithms for Elliptic Curve Cryptosystems using Endomorphisms

This book focuses on fast algorithms for computing scalar multiplication (or point multiplication) on certain types of elliptic curves, as scalar multiplication is the most time-consuming operation in elliptic curve cryptography (ECC). More precisely, the text provides readers with both a theoretical perspective on the use of low Hamming weight Frobenius expansion for scalar multiplication and a practical perspective on the implementation of scalar multiplication using the above technique. ECC has a wide range of applications, including public-key encryption and digital signatures. However, along with the use of ECC in low-end devices, the goal is to improve the efficiency of the operation. The results of this book can be used for the efficient implementation of various ECC-based applications. The book will be of interest to all readers who have at least a basic grasp of the theory of elliptic curves, or are familiar with the use of cryptography. After reading this book, readers will understand both the theory and implementation of fast scalar multiplication algorithms.

Introduction.- Mathematical Background.- Groups, Rings, Fields.- Elliptic Curves.- Scalar Multiplication.- Fast Computation Methods of Scalar Multiplication.- Efficient Arithmetic on Subfield Elliptic Curves Over Small Finite Fields of Odd Characteristic.- Explicit Lower Bound for the Length of Minimal Weight Expansions on Koblitz Curves.- Lower Bound for the Length of Minimal Weight Expansions on Koblitz Curves.- Another Proof of the Minimality of the Hamming Weight.- Minimal Length Form on Koblitz Curves and its Cryptographic Application.