More Efficient $k$-wise Independent Permutations from Random Reversible Circuits via log-Sobolev Inequalities
<|reference_start|>More Efficient $k$-wise Independent Permutations from Random Reversible Circuits via log-Sobolev Inequalities: We prove that the permutation computed by a reversible circuit with $\tilde{O}(nk\cdot \log(1/\varepsilon))$ random $3$-bit gates is $\varepsilon$-approximately $k$-wise independent. Our bound improves on currently known bounds in the regime when the approximation error $\varepsilon$ is not too small. We obtain our results by analyzing the log-Sobolev constants of appropriate Markov chains rather than their spectral gaps.<|reference_end|>
@article{gretta2024more,
    title={More Efficient $k$-wise Independent Permutations from Random Reversible
  Circuits via log-Sobolev Inequalities},
    author={Lucas Gretta, William He, Angelos Pelecanos},
    journal={arXiv preprint arXiv:2406.08499},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.08499},
    primaryClass={cs.CC cs.CR}
}