An Introduction to the Theory of Elliptic Curves

An Introduction to theTheory of Elliptic CurvesJoseph H. SilvermanBrown University andNTRU Cryptosystems, School onComputational Number Theory andApplications to CryptographyUniversity of WyomingJune 19 July 7, 20060An Introduction to the Theory of Elliptic CurvesOutline Introduction Elliptic Curves The Geometry of Elliptic Curves The Algebra of Elliptic Curves What DoesE(K) Look Like? Elliptic Curves Over Finite Fields The Elliptic Curve Discrete Logarithm Problem reduction Modulop, Lifting, and Height Functions Canonical Heights on Elliptic Curves Factorization Using Elliptic Curves L-Series, birch Swinnerton-Dyer, and $1,000,000 Additional Material Further ReadingAn Introduction to the Theory of Elliptic Curves 1 An Introduction to the Theory of Elliptic CurvesThe Discrete Logarithm ProblemFix a groupGand an elementg G. TheDiscreteLogarithm Problem(DLP) forGis:Given an elementhin the subgroupgenerated byg, find an integermsatisfyingh= smallest integermsatisfyingh=gmis called thelogarithm(orindex) ofhwith respect tog, and isdenotedm= logg(h) orm= indg(h).

† Reduction Modulo p, Lifting, and Height Functions † Canonical Heights on Elliptic Curves † Factorization Using Elliptic Curves † L-Series, Birch{Swinnerton-Dyer, and $1,000,000 † Additional Material † Further Reading An Introduction to the Theory of Elliptic Curves { 1

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  Introduction, Reduction, Theory, Curves, Birch, Elliptic, An introduction to the theory of elliptic curves

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