Average-Case Hardness of Parity Problems: Orthogonal Vectors, k-SUM and More

This work establishes conditional lower bounds for average-case {\em parity}-counting versions of the problems $k$-XOR, $k$-SUM, and $k$-OV. The main contribution is a set of self-reductions for the problems, providing the first specific distributions, for which: $\mathsf{p}\mathsf{a}\mathsf{r}\mathsf{i}\mathsf{t}\mathsf{y}\text{-}k\text{-}OV$ is $n^{\mathrm{\Omega }(\sqrt{k})}$ average-case hard, under the $k$-OV hypothesis (and hence under SETH), $\mathsf{p}\mathsf{a}\mathsf{r}\mathsf{i}\mathsf{t}\mathsf{y}\text{-}k\text{-}SUM$ is $n^{\mathrm{\Omega }(\sqrt{k})}$ average-case hard, under the $k$-SUM hypothesis, and $\mathsf{p}\mathsf{a}\mathsf{r}\mathsf{i}\mathsf{t}\mathsf{y}\text{-}k\text{-}XOR$ is $n^{\mathrm{\Omega }(\sqrt{k})}$ average-case hard, under the $k$-XOR hypothesis. Under the very believable hypothesis that at least one of the $k$-OV, $k$-SUM, $k$-XOR or $k$-Clique hypotheses is true, we show that parity-$k$-XOR, parity-$k$-SUM, and parity-$k$-OV all require at least $n^{\mathrm{\Omega }(k^{1\mathrm{/}3})}$ (and sometimes even more) time on average (for specific distributions). To achieve these results, we present a novel and improved framework for worst-case to average-case fine-grained reductions, building on the work of Dalirooyfard, Lincoln, and Vassilevska Williams, FOCS 2020.