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Mathematical Foundations of Elliptic Curve Cryptography (PDF 113P)
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 
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, Abel’s theorem, The j-invariant, 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, Abel’s theorem and Riemann-Roch, Weierstrass normal forms, The j-invariant, Arithmetic of elliptic curves, Rational points on elliptic curves, Reduction modulo primes and torsion points, An intermezzo on group cohomology, The weak Mordell-Weil theorem, Heights and the Mordell-Weil theorem.
Author(s): Thomas Kramer
77 Pages
Elliptic Curve Cryptography by Abhijit Das
This note explains the following topics: Arithmetic of Elliptic Curves, Classical Elliptic-Curve Cryptography, Efficient Implementation, Introduction to Pairing, Pairing-Based Cryptography, Sample Application—ECDSA 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, Landen’s Transformation, Complete Functions, Development of Elliptic Functions into Factors, Elliptic Integrals of the Second Order, Numerical Calculations.
Author(s): Arthur L. Bake
147 Pages