# time complexity and big O notation of sub set dynamic programming

given set of $$n$$ positive integers and a target number $$T$$ there is an dynamic programming algorithm that run in $$O(n T)$$ time complexity that solves the sub-set sum problem, It is regarded as exponential in terms of $$T$$, when $$T$$ is a big number, for the sake of this question assume that there is an algorithm that solves sub-set sum in $$O(n \ln {T})$$ time complexity, does the algorithm still runs in exponential time and why ? or does it runs in polynomial time(that will put sub-set sum in P) and why ?

I am leaning toward the polynomial time algorithm since you need $$O(\ln {T})$$ time at least for writing $$T$$ digits in the memory !! I might be very wrong !!

The standard algorithm with time $$O(nT)$$ isn't exponential in terms of $$T$$, it's exponential in terms of input length, as instance with input $$T$$ (and $$n$$ numbers $$\leq T$$ each) has length $$O(n\log T)$$.
The theoretical algorithm with complexity $$O(n \log T)$$ would run in polynomial time of input size.