On the Hardness of Average-case k-SUM

In this work, we show the first worst-case to average-case reduction for the classical $k$-SUM problem. A $k$-SUM instance is a collection of $m$ integers, and the goal of the $k$-SUM problem is to find a subset of $k$ elements that sums to $0$. In the average-case version, the $m$ elements are chosen uniformly at random from some interval $[-u,u]$. We consider the total setting where $m$ is sufficiently large (with respect to $u$ and $k$), so that we are guaranteed (with high probability) that solutions must exist. Much of the appeal of $k$-SUM, in particular connections to problems in computational geometry, extends to the total setting. The best known algorithm in the average-case total setting is due to Wagner (following the approach of Blum-Kalai-Wasserman), and achieves a run-time of $u^{O(1/\log k)}$. This beats the known (conditional) lower bounds for worst-case $k$-SUM, raising the natural question of whether it can be improved even further. However, in this work, we show a matching average-case lower-bound, by showing a reduction from worst-case lattice problems, thus introducing a new family of techniques into the field of fine-grained complexity. In particular, we show that any algorithm solving average-case $k$-SUM on $m$ elements in time $u^{o(1/\log k)}$ will give a super-polynomial improvement in the complexity of algorithms for lattice problems.