Nemotron-IMO-Bench 2026 N20

Let $n>1$ be an integer. For which functions $f: \mathbf{Z}\to \{1,2,...,n\}$ is there a function $j: \{1,2,...,n-1\}\to \mathbf{Z}$ such that for each $k\in \{1,2,...,n-1\}$ and $m\in \mathbf{Z}$ $$f(m+j(k))\equiv f(m+k)-f(m)\pmod {n+1}?$$