Joint Sparse Form of Window Three for Koblitz Curve

The joint sparse form (JSF) for the non-adjacent form (NAF) representation of two large integers a and b, was proposed by Solinas. Then Ciet extended it to the φ-JSF for the φ-NAF representations of a and b using the endomorphism φ when computing aP+bQ , where P andQ are two points on the elliptic curve, in elliptic curve cryptography (ECC). It can be observed that τ -JSF is a special case of φ-JSF. In this paper, we will extend the τ -JSF idea to window 3 (RTNAF3), referred to as window three τ joint sparse form (WTT-JSF). Mathematical analysis shows that a number of additions can be eliminated with this representation. Moreover, a detail derivation of the length and density of this form is given. The density is 11/27 which is lower than 7/16 when RTNAF3 is applied directly.