Talk to Groningen on isogenous hyperelliptic curves of genus 2

Talk to Groningen, 13-11-2009 11:45-12:45

Marios Magioladitis,

The discrete logarithm problem on isogenous

hyperelliptic curves of genus 2

Abstract. In 2005, Jao, Miller, and Venkatesan proved that the DLP of elliptic curves with the same endomorhism ring is random reducible under the GRH. In this talk, we discuss a possible generalization of this result to hyperelliptic curves of genus 2 (and 3) defined over a finite field and show the difficulties involved. First, we explain the role of the endomorphism rings of the Jacobian and the polarization. Following the work of Jao, Miller and Venkatesan, we construct isogeny graphs for genus 2 curves. Specifically, we discuss the connection between isogenies and ideal classes in the Jacobian of these curves. This project is research in progress and we describe the current status of this research.