Elliptic nets and elliptic curves

The sequence of division polynomials for an elliptic curve satisfies a non-linear recurrence relation. Specialising to a chosen elliptic curve and evaluating at a chosen point gives a recurrence sequence in the field over which curve and point are defined. In the field of rational numbers, Morgan Ward showed in 1948 that all sequences satisfying this particular recurrence relation arise from division polynomials. These are called elliptic divisibility sequences. In this thesis, we define a higher rank generalisation of elliptic divisibility sequences called elliptic nets. To do so, we define rational functions called net polynomials in analogy to division polynomials. For any integer n, we define a collection of such net polynomials in n variables indexed by n-tuples of integers; for n = 1, one obtains the division polynomials. This collection satisfies a certain non-linear recurrence relation. Any array satisfying this relation is called an elliptic net. The evaluation of the array of functions at a curve and n-tuple of points gives an elliptic net with values in K . Conversely, any elliptic net over K arises from the net polynomials evaluated at some elliptic curve and tuple of points. In this thesis, we make precise the correspondence between elliptic curves and elliptic nets, over arbitrary fields. We describe the Laurentness properties of elliptic nets, and generalise the 'symmetry properties' observed by Morgan Ward and others. It is shown that the Poincare biextension of an elliptic curve crossed with itself has a factor system given by the net polynomials. As a consequence, the Tate-Lichtenbaum and Weil pairings for an elliptic curve have a description in terms of elliptic nets. This leads to a new algorithm for computing these pairings by computing terms of elliptic nets. The complexity of this algorithm is examined. Finally, some hard computational problems for elliptic nets are related to the elliptic curve discrete logarithm problem over finite fields, with a view toward cryptographic security.