Constructions of $k$-Min-Wise Hash from Bounded Independence

Min-wise hashing and its $k$-min-wise extension are fundamental tools in sampling, sketching, and similarity estimation. A standard approach to constructing such families is bounded independence. For ordinary min-wise hashing, the required degree of independence is fully understood: $\Theta(\log 1/\delta)$-wise independence is both sufficient and necessary. For $k$-min-wise hashing, however, the best previous result only showed that $O(k\log\log1/\delta+\log1/\delta)$-wise independence suffices, with no matching lower bound. We give a tight characterization of the amount of bounded independence required for $k$-min-wise hashing, proving that $\Theta(k+\log1/\delta)$-wise independence is both sufficient and necessary. This improves the previous upper bound and provides a matching lower bound. Consequently, the standard construction of bounded-independent hash families has seed length $O\big((k+\log1/\delta)\log(N/\delta)\big)$. In particular, for any polynomially small error $\delta$ and any $k=\Omega(\log N)$, it achieves the optimal seed length $O(k\log N)$. We also study random affine hash functions over $\mathbb{F}_2$ and show that, despite being pairwise independent, they may incur multiplicative error $\Omega(\log n)$ even for ordinary min-wise hashing.