I'm looking for a problem that is NP-Complete (even) if the number of input values is at most a polylog of the input size. So some or each value in the input should be so large that the number of values is "negligable".
One way to do this is to say that there is only one input value that should be split and represents the input to another NPC problem. But I consider this a bypass and I'm looking for a more conventional problem.
According to Does the complexity of strongly NP-hard or -complete problems change when their input is unary encoded? and its answers, I cannot simply take a strongly NP-Complete problem and represent its input as unary.