Explicit Constructions of Canonical and Absolute Minimal Degree Lifts of Twisted Edwards Curves

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 formulas are given.

Additionally, we define a notion of Minimal and Absolute Minimal Degree Liftings modulo pn and an associated lift of points we call an Edwards Lift. We show how the Canonical Lift may be used to find examples of Absolute Minimal Degree Liftings, at least modulo p3.