May 2003	Jean-Yves Enjalbert	Jacobiennes et Cryptographie (Universit'e de Limoges)	Universit'e de Lille, France.
April 2007	Jennifer Paulhus	Elliptic factors in Jacobians of low genus curves	Department of Mathematics, Kansas State University, Manhattan, KS.
May 2007	Samuel Kadziela	Rigid analytic uniformization of hyperelliptic curves	Department of Mathematics, University of California, Irvine, CA.
August 2007	SeungKook Park	Applications of algebraic curves to cryptography	Department of Mathematical Sciences, University of Cincinnati, OH.

Publications by students

Decomposing Jacobians of curves with extra automorphisms, Jennifer Paulhus, to appear in Acta Arithmetica.

Rigid Analytic Uniformization of Curves and the Study of Isogenies, Samuel Kadziela, Acta Appl. Math. 99 (2007), no. 2, 185--204.

Abstract: Complex uniformization of curves is a popular tool in Number Theory. There are, however, some arithmetic and computational advantages in the use of p-adic uniformization. This paper compares the two theories and discusses how they can be used to study isogenies, with explicit examples of p-adic uniformization of hyperelliptic curves.

The Vector Decomposition Problem for Elliptic and Hyperelliptic Curves, Iwan Duursma and Negar Kiyavash, J. Ramanujan Math. Soc. 20 (2005), no. 1, 59--76. [.pdf] [.ps] (scanned version in pdf of the paper [Yos03] by Yoshida)

Abstract: The group of m-torsion points on an elliptic curve, for a prime number m, forms a two-dimensional vector space. It was suggested and proven by Yoshida that under certain conditions the vector decomposition problem (VDP) on a two-dimensional vector space is at least as hard as the computational Diffie-Hellman problem (CDHP) on a one-dimensional subspace. In this work we show that even though this assessment is true, it applies to the VDP for m-torsion points on an elliptic curve only if the curve is supersingular. But in that case the CDHP on the one-dimensional subspace has a known sub-exponential solution. Furthermore, we present a family of hyperelliptic curves of genus two that are suitable for the VDP.

A generalized floor bound for the minimum distance of geometric Goppa codes, Benjamin Lundell and Jason McCullough, J. Pure Appl. Algebra 207 (2006), no. 1, 155--164.

Abstract: We prove a new bound for the minimum distance of geometric Goppa codes that generalizes two previous improved bounds. We include examples of the bound applied to one- and two-point codes over certain Suzuki and Hermitian curves.

Geometric Reed-Solomon codes of length 64 and 65 over F8, Chien-Yu Chen and Iwan Duursma, IEEE Trans. on Inform. Theory, vol. 49, pp. 1351-1353, May 2003. [.ps]

Abstract: We determine the actual parameters for a class of one-point codes of length 64 and 65 over F_8 defined by Hansen-Stichtenoth. Several codes have a minimum distance that exceeds the Feng-Rao bound. The codes with parameters [64,5,51], [64,10,43],[64,11,42], [64,12,40], [65,5,52],[65,10,43],[65,11,42],[65,12,41],[64,13,40] are better than any known code.

Bounds for completely decomposable Jacobians, Duursma, I.M., and Enjalbert, J.-Y., Finite fields with applications to coding theory, cryptography and related areas (Oaxaca, 2001), 86--93, Springer, Berlin, 2002. [.ps]

Abstract: A curve over the field of two elements with completely decomposable Jacobian is shown to have at most six rational points and genus at most 26. The bounds are sharp. The previous upper bound for the genus was 145. We also show that a curve over the field of q elements with more than q^{m/2}+1 rational points has at least one Frobenius angle in the open interval (\pi/m,3\pi/m). The proofs make use of the explicit formula method.