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 $\tilde{O}(nk\cdot \log (1\mathrm{/}\epsilon ))$ random $3$-bit gates is $\epsilon$-approximately $k$-wise independent. Our bound improves on currently known bounds in the regime when the approximation error $\epsilon$ is not too small. We obtain our results by analyzing the log-Sobolev constants of appropriate Markov chains rather than their spectral gaps.