On the Optimal Parameter Choice for Elliptic Curve Cryptosystems Using Isogeny

The isogeny for elliptic curve cryptosystems was initially used for the efficient improvement of order counting methods. Recently, Smart proposed the countermeasure using isogeny for resisting the refined differential power analysis by Goubin (Goubin’s attack). In this paper, we examine the countermeasure using isogeny against zero-value point (ZVP) attack that is generalization of Goubin’s attack. We show that some curves require higher order of isogeny to prevent ZVP attack. Moreover, we prove that this countermeasure cannot transfer a class of curve to the efficient curve that is secure against ZVP attack. This class satisfies that the curve order is odd and (−3/p) = −1 for the base field p, and includes three SECG curves. In the addition, we compare some efficient algorithms that are secure against both Goubin’s attack and ZVP attack, and present the most efficient method of computing the scalar multiplication for each curve from SECG. Finally, we discuss another improvement for the efficient scalar multiplication, namely the usage of the point (0, y) for the base point of curve parameters. We are able to improve about 11% for double-and-add-always method, when the point (0, y) exists in the underlying curve or its isogeny.