looking for a strongly NP Complete related problem
Wikipedia's list of NP-complete problems lists the following problem:

Input: positive integers $A,B,C$
Question: do there exist positive integers $x,y$ such that $Ax^2+By-C=0$?

This is a NP-complete problem that has only 3 input values (which is certainly at most poly-log of the input size).



You can find another example of a NP-complete problem with only 3 input values at A Query regarding Quadratic Residuocity Problem.