On the Optimal Parameter Choice for Elliptic Curve Cryptosystems Using Isogeny

Toru AKISHITA  Tsuyoshi TAKAGI  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E88-A   No.1   pp.140-146
Publication Date: 2005/01/01
Online ISSN: 
DOI: 10.1093/ietfec/e88-a.1.140
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)
Category: Tamper-Resistance
Keyword: 
elliptic curve cryptosystems,  isomorphism,  isogeny,  side channel attack,  zero-value point attack,  

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Summary: 
Isogeny for elliptic curve cryptosystems was initially used for efficient improvement of order counting methods. Recently, Smart proposed a countermeasure using isogeny for resisting a refined differential power analysis by Goubin (Goubin's attack). In this paper, we examine a 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 the class of curves that satisfies (-3/p) = 1 and whose order is odd cannot be mapped by isogeny to curves with a = -3 and secure against ZVP attack. We point out that three SECG curves are in this class. 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 a scalar multiplication for each curve from SECG. Finally, we discuss another improvement for an efficient scalar multiplication, namely the usage of a point (0,y) for a 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 an underlying curve or its isogeny.