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, including conditional lower bounds for $r^k \leq M \leq r^{2k}$, a search-to-decision reduction for $M > r^k$, hardness amplification for $M \geq r^k$, a construction of PKE for some $M \leq 2^{\mathrm{polylog}(r)}$, and non-trivial algorithms for any $M \geq 2^{r^2}$.

翻译：在平均情形$k$-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$k$个数使其模$M$的和为$0$（这组数称为一个解）。在相关的$k$-XOR问题中，给定$k$个长度为$\log{M}$的均匀随机布尔向量，目标是找到其中$k$个向量使得它们的按位异或结果为全零向量。这两个问题在细粒度复杂度分析与密码分析中具有广泛应用。这些问题的可行性及复杂度取决于$k$、$r$和$M$的相对取值。在稠密情形（$M \leq r^k$）中，解以高概率存在，该情形已被充分理解，并存在若干非平凡算法和难度猜想。而在稀疏情形（$M\gg r^k$）中，解几乎不可能存在，我们对许多基本问题的最佳认知仍局限于从稠密或最坏情形中直接迁移的结果。我们研究了稀疏情形下的种植$k$-SUM和$k$-XOR问题。在这类问题中，随机生成的实例中会植入一个随机解，需要将其恢复。当$M$超过$r^k$时，这些植入解往往以递增的概率成为唯一解，从而可能更易被找到。我们展示了关于这些问题复杂度与应用的若干结果，包括：当$r^k \leq M \leq r^{2k}$时的条件性下界、当$M > r^k$时的搜索到判定归约、当$M \geq r^k$时的难度放大、对某些$M \leq 2^{\mathrm{polylog}(r)}$的基于公钥加密的构造，以及针对任意$M \geq 2^{r^2}$的非平凡算法。