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The discrete logarithm problem in Jacobians of curves of high genus $g$ overfinite 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 curveswhose degrees in $X$ and $Y$ are low with respect to the curve genus, andsuitably unbalanced. The finite base fields are arbitrary, but their sizesshould not grow too fast compared to the genus. For this family, the groupstructure can be computed in subexponential time of $L_{q^g}(1/3, O(1))$, and adiscrete logarithm computation takes subexponential time of$L_{q^g}(1/3+\epsilon, o(1))$ for any positive $\epsilon$. These runtime boundsrely on heuristics similar to the ones used in the number field sieve or thefunction field sieve algorithms

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