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This paper presents new explicit formulae for the point doubling , tripling and addition for ordinary Weierstraß elliptic curves with a point of order 3 and their equivalent Hessian curves over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and… (More)
In this paper, we present the generalized Huff curves that contain Huff’s model as a special case. First, it is proved that every elliptic curve with three points of order 2 is isomorphic to a generalized Huff curve. Then, the fast and explicit formulae are derived for generalized Huff curves in projective coordinates. This paper also enumerates the number… (More)
Edwards curves are an alternate model for elliptic curves, which have attracted notice in cryptography. We give exact formulas for the number of F q-isomorphism classes of Edwards curves and twisted Ed-wards curves. This answers a question recently asked by R. Farashahi and I. Shparlinski.
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… (More)
It is proved in this paper that for any point on an elliptic curve, the mean value of x-coordinates of its n-division points is the same as its x-coordinate.
In this paper, the twisted Jacobi intersections which contains Ja-cobi intersections as a special case is introduced. We show that every elliptic curve over the prime field with three points of order 2 is iso-morphic to a twisted Jacobi intersections curve. Some fast explicit for-mulae for twisted Jacobi intersections curves in projective coordinates are… (More)
This paper presents a new model of ordinary elliptic curves with fast arithmetic over field of characteristic two. In addition, we propose two isomorphism maps between new curves and Weierstrass curves. This paper proposes new explicit addition law for new binary curves and prove the addition law corresponds to the usual addition law on Weierstrass curves.… (More)
This paper proposes new explicit formulae for the point doubling , tripling and addition on ordinary Weierstrass elliptic curves with a point of order 3 over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and tripling formulae in projective… (More)