Welcome to the resource topic for 2020/1136
Title:
On the Family of Elliptic Curves y^2=x^3+b/\mathbb{F}_p
Authors: Han Wu, Guangwu Xu
Abstract:This paper is devoted to a more precise classification of the family of curves E_b:y^2=x^3+b/\mathbb{F}_p. For prime p\equiv 1 \pmod 3, explicit formula of the number of \mathbb{F}_p-rational points on E_b is given based on the the coefficients of a (primary) decomposition of p=(c+d\omega)\overline{(c+d\omega)} in the ring \mathbb{Z}[\omega] of Eisenstein integers. More specifically, [ #E_b(\mathbb{F}_p)\in p+1-\big{\pm(d-2c),\pm(c+d), \pm(c-2d)\big}. ] The correspondence between these 6 numbers of points and the 6 isomorphism classes of the groups E_b(\mathbb{F}_p) can be efficiently determined. Each of these numbers can also be interpreted as a norm of an Eisenstein integer. For prime p\equiv 2 \pmod 3, it is known that E_b(\mathbb{F}_p)\cong \mathbb{Z}_{p+1}. Two efficiently computable isomorphisms are described within the single isomorphism class of groups for representatives E_1(\mathbb{F}_p) and E_{-3}(\mathbb{F}_p). The explicit formulas \#E_b(\mathbb{F}_p) for p\equiv 1 \pmod 3 are used in searching prime (or almost prime) order Koblitz curves over prime fields. An efficient procedure is described and analyzed. The procedure is proved to be deterministic polynomial time, assuming the Generalized Riemann Hypothesis. These formulas are also useful in using GLV method to perform fast scalar multiplications on the curve E_b. Several tools that are useful in computing cubic residues are also developed in this paper.
ePrint: https://eprint.iacr.org/2020/1136
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