Computing Discrete Logarithms in the Jacobian of High-Genus Hyperelliptic Curves over Even Characteristic Finite Fields

We describe improved versions of index-calculus algorithms for solving discrete logarithm problems in Jacobians of high-genus hyperelliptic curves defined over even characteristic fields. Our first improvement is to incorporate several ideas for the low-genus case by Gaudry and Theriault, including the large prime variant and using a smaller factor base, into the large-genus algorithm of Enge and Gaudry. We extend the analysis in [24] to our new algorithm, allowing us to predict accurately the number of random walk steps required to find all relations, and to select optimal degree bounds for the factor base. Our second improvement is the adaptation of sieving techniques from Flassenberg and Paulus, and Jacobson to our setting. The new algorithms are applied to concrete problem instances arising from the Weil descent attack methodology for solving the elliptic curve discrete logarithm problem, demonstrating significant improvements in practice

Time-memory trade-offs for index calculus in genus 3

Abstract. In this paper, we present a variant of Diem’s Õ(q) index calculus algorithm to attack the discrete logarithm problem (DLP) in Jacobians of genus 3 non-hyperelliptic curves over a finite field Fq. We implement this new variant in C++ and study the complexity in both theory and practice, making the logarithmic factors and constants hidden in the Õ-notation precise. Our variant improves the computational complexity at the cost of a moderate increase in memory consumption, but we also improve the computational complexity even when we limit the memory usage to that of Diem’s original algorithm. Finally, we examine how parallelization can help to reduce both the memory cost per computer and the running time for our algorithms