Chapter

Cryptographic Hardware and Embedded Systems - CHES 2009

Volume 5747 of the series Lecture Notes in Computer Science pp 304-316

Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

  • Nicolas MéloniAffiliated withDepartment of Electrical and Computer Engineering, University of Waterloo
  • , M. Anwar HasanAffiliated withDepartment of Electrical and Computer Engineering, University of Waterloo

Abstract

In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a modified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as \(\sum^n_{i=1}2^{b_i}3^{t_i}\) where (b i ) and (t i ) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we propose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups.

Keywords

Double-base number system Zeckendorf representation elliptic curve point scalar multiplication Yao’s algorithm