Classic subset-sum: Given an integer $T$ and a set of integers $S = \{x_1, x_2, \cdots, x_k\}$ is there a subset of $S$ that such that the sum of the elements of the subset is exactly $T$?
This problem is known to be NP-hard.
Is the following variant known or studied in literature?
Modular subset-sum: Given an integer $T$ and a set of integers $S = \{x_1, x_2, \cdots, x_k\}$ and a modulus $m \in \mathbb{Z}$ is there a subset of $S$ that such that the sum of the elements of the subset is exactly $T \pmod m$?
Does this variant have a polynomial time algorithm for solving it? References appreciated.
Edit: As @GregMartin points out in the comments, if $m$ is greater than the sum of absolute values of $x_j \in S$, then the problem reduces to classic subset-sum. Therefore, we can just consider $m_1 \cdot m_2 \cdot m_3 \cdots m_n > T > m_1 \cdot m_2 \cdot m_3 \cdots m_{n-1}$ where integer $m_j < T$ is the modulus in question. We can even assume that $m_j \sim O(\log T)$.
