We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an $\varepsilon>0$ and an algorithm which solves SAT in time $(2-\varepsilon)^{tw}|\phi|^{O(1)}$, where $tw$ is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time $n^{(1-\delta)\frac{k}{2}}$ for some $\delta>0$ and all sufficiently large $k$. We also establish an analogous result for the k-XOR problem, where integer addition is replaced by component-wise addition modulo $2$. As an application of our reduction we are able to revisit tight lower bounds on the complexity of several fundamental problems parameterized by treewidth (Independent Set, Max Cut, $k$-Coloring). Our results imply that these bounds, which were initially shown under the SETH, also hold if one assumes the k-SUM or k-XOR Hypotheses, arguably increasing our confidence in their validity.

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