sum of like indices in circular lists
For most of this answer, I discuss the decision version of the problem, in which an instance having at most one solution is given (the "promise"), and you have to decide whether it has any solutions or none.

You can reduce PARTITION to your problem (exercise). (PARTITION is the problem of determining whether a set of integers can be partitioned into two parts with equal sum.) Admittedly, this doesn't necessarily satisfy the condition that the solution be unique.

With some more work, you can (directly) reduce SAT to (the decision version of) your problem, and perhaps that can be done in such a way so that if the SAT instance has a unique solution, then so does the resulting instance of your problem. In that case, we get that the decision version is not solvable in polytime unless NP=RP, which is considered unlikely.

Note that if the decision version (more accurately, the promise version) of the problem is not solvable in polytime, then no algorithm can solve all YES instances in polytime: if it did, you could run it up to its allotted running time, and check the solution (if the algorithm stopped in time).