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Research Overview

My thesis (`Elliptic nets') concerns arithmetic geometry, specifically elliptic curves.  It has connections to the study of recurrence sequences and to elliptic curve cryptography.

Areas of interest include: elliptic divisibility sequences, elliptic curves, integer sequences, recurrence sequences, elliptic curve cryptography, pairings on elliptic curves, applications of methods of Diophantine approximation to arithmetic geometry, height functions on varieties, abc conjecture.

 

Elliptic Divisibility Sequences

An elliptic divisibility sequence is a sequence satisfying this recurrence relation:




The sequence is generated from the first few terms W1, ..., W4, and if these terms are integers, W1 = 1, and W2 | W4, then the entire sequence consists of integers.  However, they can be studied over any field, and they are intimately related to elliptic curves.  Graeme Everest maintains a website about these sequences, and a good place to go for references is the book Recurrence Sequences by Everest, van der Poorten, Shparlinski and Ward.
 

 

Elliptic Nets

I introduce elliptic nets in my thesis.  Elliptic nets generalise elliptic divisibility sequences to higher rank.  That is, they are arrays of numbers of any dimension (an example is the picture at the top right of my webpage).  They satisfy a generalisation of the recurrence relation above, given here:



They are also related to elliptic curves, and contain interesting arithmetic information about the curve and its structure as an abelian variety.  For instance, they can be used to calculate Weil and Tate pairings, and to construct generalised Jacobians of the curve, or the natural Poincare bi-extension of the curve.

A good introduction for number theorists is Section 1 of my preprint Elliptic Nets and Elliptic Curves.   For something even more concrete (with pretty examples), try the slides from Number Theory & Computability.

For cyptographers, see the slides from Elliptic Curve Cryptography 2007.  When you want more details, you can also see my article The Tate Pairing via Elliptic Nets. 

 

Tate Pairings in Cryptography

Elliptic nets can be used to compute the Tate pairing.  This algorithm differs from Miller's usual algorithm but has comparable run-time.  The original algorithm is described in my paper The Tate Pairing via Elliptic Nets and implemented for PARI/GP.  Several people have worked on other implementations.  Among these are Ben Lynn, who has implemented it in his Pairing-Based Cryptography Library; Graeme Taylor, who has implemented it for SAGE and has notes on his improvements; and Augusto Jun Devegili at University College Dublin.