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2009•Maynooth University ePrints and eTheses Archive (Maynooth University)Open access

Elliptic Curves Over Finite Fields

Sonia Balagopalan

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Abstract

This thesis provides a self-contained introduction to elliptic curves accessible\nto advanced undergraduates and graduate students in mathematics, with\nemphasis on the the theory of elliptic curves over finite fields.\nIn Chapter 1, affine and projective planes are introduced. Chapter 2\nintroduces the theory of algebraic curves and the Weierstrass Normal Form\nof a cubic curve is derived. In Chapter 3 we define derivations on arbitrary\npolynomial rings, and prove the group law for elliptic curves. Chapter 4\ndiscusses elliptic curves over finite fields and proves some results on counting\npoints.

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This thesis provides a self-contained introduction to elliptic curves accessible\nto advanced undergraduates and graduate students in mathematics, with\nemphasis on the the theory of elliptic curves over finite fields.\nIn Chapter 1, affine and projective planes are introduced. Chapter 2\nintroduces the theory of algebraic curves and the Weierstrass Normal Form\nof a cubic curve is derived. In Chapter 3 we define derivations on arbitrary\npolynomial rings, and prove the group law for elliptic curves. Chapter 4\ndiscusses elliptic curves over finite fields and proves some results on counting\npoints.

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Available abstract

This thesis provides a self-contained introduction to elliptic curves accessible\nto advanced undergraduates and graduate students in mathematics, with\nemphasis on the the theory of elliptic curves over finite fields.\nIn Chapter 1, affine and projective planes are introduced. Chapter 2\nintroduces the theory of algebraic curves and the Weierstrass Normal Form\nof a cubic curve is derived. In Chapter 3 we define derivations on arbitrary\npolynomial rings, and prove the group law for elliptic curves. Chapter 4\ndiscusses elliptic curves over finite fields and proves some results on counting\npoints.

Key concepts: Supersingular elliptic curve, Schoof's algorithm, Hessian form of an elliptic curve, Elliptic curve, Division polynomials, Edwards curve, Algebraic curve, Mathematics

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