Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}). In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

Elliptic Curve over a Local Finite Ring Rn

The goal of this chapter is to study some arithmetic proprieties of an elliptic curve defined by a Weierstrass equation on the local ring Rn=FqX/Xn, where n≥1 is an integer. It consists of, an introduction, four sections, and a conclusion. In the first section, we review some fundamental arithmetic proprieties of finite local rings Rn, which will be used in the remainder of the chapter. The second section is devoted to a study the above mentioned elliptic curve on these finite local rings for arbitrary characteristics. A restriction to some specific characteristic cases will then be considered in the third section. Using these studies, we give in the fourth section some cryptography applications, and we give in the conclusion some current research perspectives concerning the use of this kind of curves in cryptography. We can see in the conclusion of research in perspectives on these types of curves.

Elliptic curves over the ring R

Let Fq be a finite field of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted Ea,b(Fq[e]) over the ring Fq[e], where e2 = e and (a,b) ∈ (Fq[e])2. In a first time, we study the arithmetic of this ring. In addition, using the Weierstrass equation, we define the elliptic curve Ea,b(Fq[e]) and we will show that Eπ0(a),π0(b)(Fq) and Eπ1(a),π1(b)(Fq) are two elliptic curves over the field Fq, where π0 and π1 are respectively the canonical projection and the sum projection of coordinates of X ∈Fq[e]. Precisely, we give a bijection between the sets Ea,b(Fq[e]) and Eπ0(a),π0(b)(Fq)×Eπ1(a),π1(b)(Fq).

ELLIPTIC CURVE POINTS AND DIOPHANTINE MODELS OF ℤ IN LARGE SUBRINGS OF NUMBER FIELDS

Let K be a number field such that there exists an elliptic curve E of rank one over K. For a set [Formula: see text] of primes of K, let [Formula: see text]. Let P ∈ E(K) be a generator of E(K) modulo the torsion subgroup. Let (xn(P), yn(P)) be the affine coordinates of [n]P with respect to a fixed Weierstrass equation of E. We show that there exists a set [Formula: see text] of primes of K of natural density one such that in [Formula: see text] multiplication of indices (with respect to some fixed multiple of P) is existentially definable and therefore these indices can be used to construct a Diophantine model of ℤ. We also show that ℤ is definable over [Formula: see text] using just one universal quantifier. Both the construction of a Diophantine model using the indices and the first-order definition of ℤ can be lifted to the integral closure of [Formula: see text] in any infinite extension K∞ of K as long as E(K∞) is finitely generated and of rank one.

Counting models of genus one curves

Let C be a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation defining C up to isomorphism. In this paper we consider genus one equations of degree n defining C, namely a (generalised) binary quartic when n = 2, a ternary cubic when n = 3 and a pair of quaternary quadrics when n = 4. In general, minimal genus one equations of degree n are not unique up to isomorphism. We explain how the number of these equations varies according to the Kodaira symbol of the Jacobian of C. Then we count these equations up to isomorphism over a number field of class number 1.

Point Counting in Families of Hyperelliptic Curves in Characteristic 2

Let EΓ be a family of hyperelliptic curves over F2alg cl with general Weierstrass equation given over a very small field F. The author of this paper describes an algorithm for computing the zeta function of Eγ, with γ in a degree n extension field of F, which has time complexity O(n3 + ε) bit operations and memory requirements O(n2) bits. Using a slightly different algorithm, one can get time O(n2.667) and memory O(n2.5), and the computation for n curves of the family can be done in time O(n3.376). All of these algorithms are polynomial-time in the genus.

DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN ℝ4: A WAY TO FURTHER GENERALIZED WEIERSTRASS EQUATION

In the previous report (J. Phys.A30 (1997) 4019–4029), I showed that the Dirac operator defined over a conformal surface immersed in ℝ3 by means of confinement procedure is identified with the differential operator of the generalized Weierstrass equation and the Lax operator of the modified Novikov–Veselov (MNV) equation. In this article, using the same procedure, I determine the Dirac operator defined over a conformal surface immersed in ℝ4, which is for a Dirac field confined in the surface. Then it is reduced to the Lax operators of the nonlinear Schrödinger and the MNV equations by taking appropriate limits. It means that the Dirac operator is related to the further generalized Weierstrass equation for a surface in ℝ4.