Elliptic Curves by Jim Milne

Park City lectures on elliptic curves over function fields

This note contains below topics, Background on curves and function fields, Elliptic curves over function fields, Surfaces and the Tate conjecture.

Author(s): Douglas Ulmer

72 Pages

Elliptic Curves An Introduction

This note describes the following topics, Solving cubic equations in two variables, Group law on the cubic curve, Theta functions, Rank two vector bundles on elliptic curves.

Author(s): University of Limerick

99 Pages

Elliptic Curves by Thomas Kramer

Elliptic curves belong to the most fundamental objects in mathematics and connect many different research areas such as number theory, algebraic geometry and complex analysis. Their definition and basic properties can be stated in an elementary way, Roughly speaking, an elliptic curve is the set of solutions to a cubic equation in two variables over a field. This PDF covers the following topics related to Elliptic Curves, Analytic theory of elliptic curves, Elliptic integrals, The topology of elliptic curves, Elliptic curves as complex tori, Complex tori as elliptic curves, Geometric form of the group law, Abels theorem, The jinvariant, The valence formula, Geometry of elliptic curves, Affine and projective varieties, Smoothness and tangent lines, Intersection theory for plane curves, The group law on elliptic curves, Abels theorem and RiemannRoch, Weierstrass normal forms, The jinvariant, Arithmetic of elliptic curves, Rational points on elliptic curves, Reduction modulo primes and torsion points, An intermezzo on group cohomology, The weak MordellWeil theorem, Heights and the MordellWeil theorem.

Author(s): Thomas Kramer

77 Pages

Elliptic Curve Cryptography by Abhijit Das

This note explains the following topics, Arithmetic of Elliptic Curves, Classical EllipticCurve Cryptography, Efficient Implementation, Introduction to Pairing, PairingBased Cryptography, Sample ApplicationECDSA Batch Verification.

Author(s): Abhijit Das

137 Pages

Elliptic Functions An Elementary Text Book for Students of Mathematics

This note explains the following topics, Elliptic Integrals, Elliptic Functions, Periodicity of the Functions, Landens Transformation, Complete Functions, Development of Elliptic Functions into Factors, Elliptic Integrals of the Second Order, Numerical Calculations.

Author(s): Arthur L. Bake

147 Pages

Elliptic Curve Cryptography by Christian Wuthrich

Ellipticcurve cryptography is an approach to publickey cryptography based on the algebraic structure of elliptic curves over finite fields.

Author(s): Christian Wuthrich

98 Pages

Elliptic Curves by Samuele Anni

This note explains the following topics, Plane curves, Projective space and homogenisation, Rational points on curves, BachetMordell equation, Congruent number curves, Elliptic curves and group law, Integer Factorization Using Elliptic Curves, Isomorphisms and jinvariant, Elliptic curves over C, Endomorphisms of elliptic curves, Elliptic Curves over finite fields, The MordellWeil Theorem, Elliptic Curve Cryptography.

Author(s): Samuele Anni

126 Pages

Mathematical Foundations of Elliptic Curve Cryptography

This note covers the following topics, algebraic curves, elliptic curves, elliptic curves over special fields, more on elliptic divisibility sequences and elliptic nets, elliptic curve cryptography, computational aspects, elliptic curve discrete logarithm.

Author(s): Prof. Dipl-Ing, Dr. techn. Michael Drmota

113 Pages