An average-case variant of the $k$-SUM conjecture asserts that finding $k$ numbers that sum to 0 in a list of $r$ random numbers, each of the order $r^k$, cannot be done in much less than $r^{\lceil k/2 \rceil}$ time. On the other hand, in the dense regime of parameters, where the list contains more numbers and many solutions exist, the complexity of finding one of them can be significantly improved by Wagner's $k$-tree algorithm. Such algorithms for $k$-SUM in the dense regime have many applications, notably in cryptanalysis. In this paper, assuming the average-case $k$-SUM conjecture, we prove that known algorithms are essentially optimal for $k= 3,4,5$. For $k>5$, we prove the optimality of the $k$-tree algorithm for a limited range of parameters. We also prove similar results for $k$-XOR, where the sum is replaced with exclusive or. Our results are obtained by a self-reduction that, given an instance of $k$-SUM which has a few solutions, produces from it many instances in the dense regime. We solve each of these instances using the dense $k$-SUM oracle, and hope that a solution to a dense instance also solves the original problem. We deal with potentially malicious oracles (that repeatedly output correlated useless solutions) by an obfuscation process that adds noise to the dense instances. Using discrete Fourier analysis, we show that the obfuscation eliminates correlations among the oracle's solutions, even though its inputs are highly correlated.

翻译：k-SUM猜想的平均情况变体断言：在一个包含r个随机数（每个数阶为r^k）的列表中，寻找k个和为0的数不能在远少于r^{⌈k/2⌉}的时间内完成。另一方面，在参数密集的情况下，当列表包含更多数字且存在多个解时，寻找其中一个解的复杂度可通过Wagner的k树算法显著提升。此类密集情况下的k-SUM算法具有众多应用，尤其在密码分析领域。本文假设平均情况k-SUM猜想成立，证明对于k=3,4,5，已知算法本质上是最优的；对于k>5，我们在有限参数范围内证明了k树算法的最优性。我们还针对将求和替换为异或运算的k-XOR问题证明了类似结果。我们的结论通过自归约方法得到：给定一个解较少的k-SUM实例，我们从中生成多个密集情况下的实例。通过利用密集k-SUM预言机求解每个实例，并期望某个密集实例的解也能解决原始问题。针对可能恶意的预言机（即反复输出相关无用解的问题），我们采用向密集实例添加噪声的混淆过程进行处理。利用离散傅里叶分析，我们证明即使输入高度相关，混淆仍能消除预言机输出解之间的相关性。