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Full-Text Articles in Number Theory
Explicit Constructions Of Canonical And Absolute Minimal Degree Lifts Of Twisted Edwards Curves, William Coleman Bitting Iv
Explicit Constructions Of Canonical And Absolute Minimal Degree Lifts Of Twisted Edwards Curves, William Coleman Bitting Iv
Doctoral Dissertations
Twisted Edwards Curves are a representation of Elliptic Curves given by the solutions of bx^2 + y^2 = 1 + ax^2y^2. Due to their simple and unified formulas for adding distinct points and doubling, Twisted Edwards Curves have found extensive applications in fields such as cryptography. In this thesis, we study the Canonical Liftings of Twisted Edwards Curves and the associated lift of points Elliptic Teichmu ̈ller Lift. The coordinate functions of the latter are proved to be polynomials, and their degrees and derivatives are computed. Moreover, an algorithm is described for explicit computations, and some properties of the general …
Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein
Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein
Doctoral Dissertations
Given an ordinary elliptic curve E over a field 𝕜 of characteristic p, there is an elliptic curve E over the Witt vectors W(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over 𝔽_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on …
Elliptic Curves Over Finite Fields, Christopher S. Calger
Elliptic Curves Over Finite Fields, Christopher S. Calger
Honors Theses
The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of Fq-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …