In the average-case $k$-SUM problem, given $r$ integers chosen uniformly at random from $\{0,\ldots,M-1\}$, the objective is to find a set of $k$ numbers that sum to $0$ modulo $M$ (this set is called a solution). In the related $k$-XOR problem, given $k$ uniformly random Boolean vectors of length $\log{M}$, the objective is to find a set of $k$ of them whose bitwise-XOR is the all-zero vector. Both of these problems have widespread applications in the study of fine-grained complexity and cryptanalysis. The feasibility and complexity of these problems depends on the relative values of $k$, $r$, and $M$. The dense regime of $M \leq r^k$, where solutions exist with high probability, is quite well-understood and we have several non-trivial algorithms and hardness conjectures here. Much less is known about the sparse regime of $M\gg r^k$, where solutions are unlikely to exist. The best answers we have for many fundamental questions here are limited to whatever carries over from the dense or worst-case settings. We study the planted $k$-SUM and $k$-XOR problems in the sparse regime. In these problems, a random solution is planted in a randomly generated instance and has to be recovered. As $M$ increases past $r^k$, these planted solutions tend to be the only solutions with increasing probability, potentially becoming easier to find. We show several results about the complexity and applications of these problems.

翻译：在平均情况下的$k$-SUM问题中，给定从$\{0,\ldots,M-1\}$中均匀随机选取的$r$个整数，目标是找到一组和为$0$模$M$的$k$个数（这组数称为解）。在相关的$k$-XOR问题中，给定$k$个均匀随机的长度为$\log M$的布尔向量，目标是找到其中$k$个向量，使其按位XOR结果为零向量。这两个问题在细粒度复杂度分析和密码分析中具有广泛应用。这些问题的可行性和复杂性取决于$k$、$r$和$M$的相对取值。在$M \leq r^k$的密集区域（解以高概率存在）已被充分理解，我们已有若干非平凡算法和难度猜想。而在$M\gg r^k$的稀疏区域（解不太可能存在）中，我们了解甚少。对于许多基本问题，目前最佳答案仅限于从密集或最坏情况设置中继承的结果。我们研究稀疏区域中的植选$k$-SUM和$k$-XOR问题。在这些问题中，随机生成的实例中植入了随机解，需要将其恢复。随着$M$超过$r^k$，这些植选解往往以越来越高的概率成为唯一解，可能变得更容易发现。我们展示了关于这些问题复杂性和应用的若干结果。