An $L (1/3)$ Discrete Logarithm Algorithm for Low Degree Curves

TL;DR

This paper introduces a subexponential algorithm for solving discrete logarithms in Jacobians of low-degree plane curves over finite fields, extending the applicability of index calculus methods to new curve families.

Contribution

The authors develop a novel $L(1/3)$ algorithm tailored for Jacobians of low-degree plane curves, expanding the scope of discrete logarithm computations in algebraic geometry.

Findings

Algorithm achieves subexponential time $L_{q^g}(1/3, O(1))$

Applicable to families of curves with low degree relative to genus

Runtime relies on heuristics similar to number field and function field sieves

Abstract

We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in $X$ and $Y$ are low with respect to their genera. The finite base fields $\FF_q$ are arbitrary, but their sizes should not grow too fast compared to the genus. For such families, the group structure and discrete logarithms can be computed in subexponential time of $L_{q^g}(1/3, O(1))$. The runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve.