Logarithmic density of rank ≥ 1 and rank ≥ 2 genus-2 Jacobians and applications to hyperelliptic curve cryptography

In this talk we present quantitative existence results for genus-$2$ curves over $\mathbb{Q}$ whose Jacobians have Mordell--Weil rank at least $1$ or $2$, ordering the curves by the naive height of their integral Weierstrass models. If the number of curves of a given height is $X$ and the number of curves having a given rank $r$ is $X^c$ we say that the logarithmic density of rank $r$ curves is $c$. Using a geometric argument we show that the rank-$2$ genus-$2$ curves have logarithmic density $\geq 5/7$. For comparison the conjectured logarithmic density of rank-$2$ elliptic curves is $19/24$, which is less than $5/7$. We continue with results about the logarithmic densities of the quadratic twists of a genus-$2$ curve, which has consequences in a new line of quantum algorithms for the discrete logarithm problem, which was initiated by Regev.

Based on joint work with M. Barcau, V. Pasol and G. Turcas.