Fast scalar multiplication algorithms for elliptic curve cryptosystems

Abstract

Over the past few decades, the rapid development of information technology simulates the need to develop fast and secure cryptographic algorithms. In 1976, the concept of public key cryptography proposed by Diffie and Hellman revolutionized the way of secure communication by allowing the free distribution of encryption keys. In particular, elliptic curve cryptography (ECC) is a public key cryptographic scheme with great potential. It was first proposed by Koblitz and Miller independently in 1985. Since then, a lot of studies have been carried out for improving the efficiency of elliptic curve cryptosystems. An essential operation in elliptic curve cryptosystems is called scalar multiplication kP, where k is a large integer and P is a point on the elliptic curve. Its fast implementation is crucial to the efficiency of an elliptic curve cryptosystem. There has been extensive research on accelerating the kP operation and various fast algorithms have been proposed. Traditionally, the scalar k is represented in binary form and kP is computed using the ‘left-to-right” “double and add” strategy. In order to have a faster computation, the non-adjacent form (NAF), the joint sparse form (JSF) and the reduced τ-adic non-adjacent form (RTNAF) were proposed afterward. Recently, a ternary / binary approach was suggested which make use of efficient triple (3P) and double (2P) of the point P to achieve fast scalar multiplication. A similar approach uses a double-base chain to represent the scalar k was also proposed. On the other hand, point halving operation was proved to be a faster operation than point doubling and so it is beneficial to replace point doubling by point halving. In this thesis, a novel scalar multiplication algorithm based on double-base chain and point halving is proposed. Our approach can be applied to general random curves and is suitable for both software and hardware implementation. It uses new double-base chain representation for the scalar k with mixed powers of ½ and 3 instead of the traditional mixed powers of 2 and 3 is suggested. The major advantage of this form of new chain is that point doubling and quadrupling in traditional scalar multiplication can be replaced by the faster point halving operation. Experimental results show that our approach leads to a lower computational complexity which contributes to the efficient implementation of elliptic curve cryptosystems. In particular, it requires fewer number of curve operations when compared to the original double-base chain approach. Finally, further research directions are suggested and a conclusion is drawn.