We present an algorithm based on the birthday paradox, which is a low-memory parallel counterpart to the algorithm of Matsuo, Chao and Tsujii. This algorithm computes the group order of the Ja-cobian of a genus 2 curve over a finite field for which the characteristic polynomial of the Probenius endomorphism is known modulo some integer. The main tool is a 2-dimensional pseudo-random walk that allows to heuristically choose random elements in a 2-dimensional space. We analyze the expected running time based on heuristics that we validate by computer experiments. Compared with the original algorithm by Matsuo, Chao and Tsujii, we lose a factor of about 3 in running time, but the memory requirement drops from several GB to almost nothing. Our method is general and can be applied in other contexts to transform a baby-step giant-step approach into a low memory algorithm.
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