An $L (1/3 + \epsilon)$ Algorithm for the Discrete Logarithm Problem for Low Degree Curves

TL;DR

This paper introduces a new algorithm for the discrete logarithm problem in Jacobians of certain low-degree, unbalanced plane curves, achieving subexponential complexity of roughly $L_{q^g}(1/3 + ext{small})$, under heuristic assumptions.

Contribution

It presents a novel $L(1/3 + ext{small})$ algorithm for discrete logarithms on a specific family of low-degree, unbalanced curves, extending the applicability of subexponential methods.

Findings

Group structure can be computed in subexponential time $L_{q^g}(1/3, O(1))$.

Discrete logarithm computation takes subexponential time $L_{q^g}(1/3+ ext{small}, o(1))$.

Runtime bounds rely on heuristics similar to number field and function field sieve algorithms.

Abstract

The discrete logarithm problem in Jacobians of curves of high genus $g$ over finite fields $\FF_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose degrees in $X$ and $Y$ are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of $L_{q^g}(1/3, O(1))$, and a discrete logarithm computation takes subexponential time of $L_{q^g}(1/3+\epsilon, o(1))$ for any positive $\epsilon$. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.