How hard is to find the operators of an addition knowing the sum of them? I want to learn whether or no there is a cryptographic primitive,scheme assumption that is based on the following hard problem if it is hard . By hard we mean that we have a polynomial adversary: The attacker obtains a number  $\sigma$ . In order to reverse engineer that or to go one step further for the cryptanalysis she needs to break that number in a set of numbers $m_{1}, m_{2}, m_{3}, \ldots, m_{n} : \sum_{i=1}^{n}m_{i}=\sigma$ . How hard is that problem?
Well, the problem "given a finite set $A$ of integers, is there a subset that sums to a target value $B$" is known as the Subset Sum problem; it is known to be hard.  Specifically, the decisional problem (is there such a sum) is NP-complete, and the computational problem (find the subset) is NP-hard.

That means that if you could solve large instances of this problem quickly, you could use that to solve a lot of interesting problems quickly, including just about any problem in crypto (finding AES keys given plaintext/ciphertext, factoring numbers, etc).

Because of this, assuming this is a hard problem would appear to be a fairly safe assumption.

On the other hand, it's not at all clear problem your problem statement that this is the problem you're relying on.  You are you "add each index of them and construct a table"; what do you mean $n_1[0]$?  If this is bit 0 of message digest $n_0$ and you're adding all the bits 0's from the message digests to form bit 0 of $T$, then you're not relying on subset problem at all; you're relying on the related problem that uses bitwise exclusive or -- that problem is known to be easy.