An $L(1\mathrm{/}3)$ Discrete Logarithm Algorithm for Low Degree Curves

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 ${\text{FF}}_q$ 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 $L_{q^g}(1\mathrm{/}3,O(1))$. The runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve.