CLASSIFICATION OF ELEMENTS IN ELLIPTIC CURVE OVER THE RING F
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Date
2022
Authors
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Journal ISSN
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Publisher
Université de M'sila
Abstract
Let Fq["] := Fq[X]/(X4 − X3) be a finite quotient ring where "4 =
"3, with Fq is a finite field of order q such that q is a power of a prime
number p greater than or equal to 5. In this work, we will study the elliptic
curve over Fq["], "4 = "3 of characteristic p 6= 2, 3 given by homogeneous
Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b
are parameters taken in Fq["]. Firstly, we study the arithmetic operation of
this ring. In addition, we define the elliptic curve Ea,b(Fq["]) and we will
show that E 0(a), 0(b)(Fq) and E 1(a), 1(b)(Fq) are two elliptic curves over
the finite field Fq, such that 0 is a canonical projection and 1 is a sum
projection of coordinate of element in Fq["]. Precisely, we give a classification
of elements in elliptic curve over the finite ring Fq["].
Description
Keywords
elliptic curves, finite ring, finite field, projective space. 2010Mathematics Subject Classification: 14H52, 11T55, 20K30, 20K27