  In the kSUM problem we are given an array of numbers $a_1,a_2,...,a_n$ and we
are required to determine if there are $k$ different elements in this array
such that their sum is 0. This problem is a parameterized version of the
well-studied SUBSET-SUM problem, and a special case is the 3SUM problem that is
extensively used for proving conditional hardness. Several works investigated
the interplay between time and space in the context of SUBSET-SUM. Recently,
improved time-space tradeoffs were proven for kSUM using both randomized and
deterministic algorithms.
  In this paper we obtain an improvement over the best known results for the
time-space tradeoff for kSUM. A major ingredient in achieving these results is
a general self-reduction from kSUM to mSUM where $m<k$, and several useful
observations that enable this reduction and its implications. The main results
we prove in this paper include the following: (i) The best known Las Vegas
solution to kSUM running in approximately $O(n^{k-\delta\sqrt{2k}})$ time and
using $O(n^{\delta})$ space, for $0 \leq \delta \leq 1$. (ii) The best known
deterministic solution to kSUM running in approximately
$O(n^{k-\delta\sqrt{k}})$ time and using $O(n^{\delta})$ space, for $0 \leq
\delta \leq 1$. (iii) A space-time tradeoff for solving kSUM using
$O(n^{\delta})$ space, for $\delta>1$. (iv) An algorithm for 6SUM running in
$O(n^4)$ time using just $O(n^{2/3})$ space. (v) A solution to 3SUM on random
input using $O(n^2)$ time and $O(n^{1/3})$ space, under the assumption of a
random read-only access to random bits.
