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

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