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Families of Elliptic Curves with Rational 3-torsion



Dustin Moody, Hongfeng Wu


In this paper we look at three families of elliptic curves with rational 3-torsion over a finite field. These families include Hessian curves, twisted Hessian curves, and a new family we call generalized DIK curves. We find the number of Fq-isogeny classes of each family, as well as the number of Fq-isomorphism classes of the generalized DIK curves. We also include some formulas for efficient computation on these curves, improving upon known results. In particular, we find better formulas for doubling and addition on the original tripling-oriented DIK curves and also for addition and tripling on elliptic curves with j-invariant 0.
Journal of Mathematical Cryptology


Elliptic curves, Hessian curves, cryptography


Moody, D. and Wu, H. (2012), Families of Elliptic Curves with Rational 3-torsion, Journal of Mathematical Cryptology, [online], (Accessed July 14, 2024)


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Created January 30, 2012, Updated November 10, 2018