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: $\mathsf{parity}\text{-}k\text{-}OV$ is $n^{\Omega(\sqrt{k})}$ average-case hard, under the $k$-OV hypothesis (and hence under SETH), $\mathsf{parity}\text{-}k\text{-}SUM$ is $n^{\Omega(\sqrt{k})}$ average-case hard, under the $k$-SUM hypothesis, and $\mathsf{parity}\text{-}k\text{-}XOR$ is $n^{\Omega(\sqrt{k})}$ 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 $n^{\Omega(k^{1/3})}$ (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.

翻译：本研究为$k$-XOR、$k$-SUM和$k$-OV问题的平均情况{\em 奇偶性}计数版本建立了条件性下界。主要贡献在于为这些问题提出了一系列自归约方法，首次构造了满足以下条件的特定分布：在$k$-OV假设（进而在SETH）下，$\mathsf{parity}\text{-}k\text{-}OV$具有$n^{\Omega(\sqrt{k})}$的平均情况困难性；在$k$-SUM假设下，$\mathsf{parity}\text{-}k\text{-}SUM$具有$n^{\Omega(\sqrt{k})}$的平均情况困难性；在$k$-XOR假设下，$\mathsf{parity}\text{-}k\text{-}XOR$具有$n^{\Omega(\sqrt{k})}$的平均情况困难性。基于$k$-OV、$k$-SUM、$k$-XOR或$k$-Clique假设中至少有一个成立的合理前提，我们证明parity-$k$-XOR、parity-$k$-SUM和parity-$k$-OV在特定分布下均需要至少$n^{\Omega(k^{1/3})}$（有时甚至更高）的平均计算时间。为实现这些结果，我们在Dalirooyfard、Lincoln与Vassilevska Williams（FOCS 2020）研究的基础上，提出了一种新颖且改进的最坏情况到平均情况细粒度归约框架。