Shorter addition chain for smooth integers using decomposition method.

• Main Authors: Mohamed, M. A., Md. Said, Mohamad Rushdan, Mohd Atan, Kamel Ariffin, Ahmad Zulkarnain, Zuriati
• Format: Article
• Language: English; English
• Published: Taylor & Francis 2011
• Online Access: http://psasir.upm.edu.my/id/eprint/25154/1/Shorter%20addition%20chain%20for%20smooth%20integers%20using%20decomposition%20method.pdf; http://psasir.upm.edu.my/id/eprint/25154/; http://www.tandf.co.uk/journals/default.asp

An efficient computation of scalar multiplication in elliptic curve cryptography can be achieved by reducing the original problem into a chain of additions and doublings. Finding the shortest addition chain is an NP-problem. To produce the nearest possible shortest chain, various methods were introduced and most of them depends on the representation of a positive integer n into a binary form. Our method works out the given n by twice decomposition, first into its prime powers and second, for each prime into a series of 2's from which a set of rules based on addition and doubling is defined. Since prime factorization is computationally a hard problem, this method is only suitable for smooth integers. As an alternative, the need to decompose n can be avoided by choosing n of the form p1 e1p2 e2⋯r er. This shall not compromise the security of ECC since its does not depend on prime factorization problem. The result shows a significant improvement over existing methods especially when n grows very large.