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 and Q 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 (RTNAF 3 ), referred to as window

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... ee 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 RTNAF 3 is applied directly.