Moment subset sums over finite fields

The $k$-subset sum problem over finite fields is a classical NP-complete problem. Motivated by coding theory applications, a more complex problem is the higher $m$-th moment $k$-subset sum problem over finite fields. We show that there is a deterministic polynomial time algorithm for the $m$-th moment $k$-subset sum problem over finite fields for each fixed $m$ when the evaluation set is the image set of a monomial or Dickson polynomial of any degree $n$. In the classical case $m=1$, this recovers previous results of Nguyen-Wang (the case $m=1$, $p>2$) [24] and the results of Choe-Choe (the case $m=1$, $p=2$) [3].