Complexity of Membership-Testing for finite abelian groups
Verifying whether $b \in \langle h_1, \ldots, h_n\rangle$ (where $h_i$ are vectors according to the OP's comments) is equivalent to verifying whether there is a solution to this system:
$$
\begin{pmatrix}
h_{1}(1) &amp; \cdots &amp; h_{n}(1) &amp; d_1^{e_1}      &amp; 0 &amp; \cdots &amp; 0 \\\\
\vdots  &amp; \ddots &amp; \vdots  &amp; \vdots &amp; \vdots  &amp; \vdots &amp; \vdots  \\\\
h_{1}(m) &amp; \cdots &amp; h_{n}(m) &amp; 0      &amp; 0 &amp; \cdots &amp; d_m^{e_N}
\end{pmatrix}
\begin{pmatrix}
x(1) \\\\
\vdots \\\\
x(n) \\\\
y(1) \\\\
\vdots \\\\
y(m)
\end{pmatrix}
\equiv
\begin{pmatrix}
b(1) \\\\
\vdots \\\\
b(m)
\end{pmatrix}
$$

In your case $e_1, \ldots, e_N$ are tiny numbers (ie. their value is logartihmic in the input size). Unfortunately, it doesn't seem like we can assume that $d_1, \ldots, d_n$ are tiny.

If they are, then you can find a solution to the system in $\text{NC}^3$ from a result of McKenzie &amp; Cook [1]. This paper shows that solving linear congruences modulo an integer with tiny factors (LCON) is in $\text{NC}^3$. Moreover, this problem is $\text{NC}^1$-equivalent to the abelian permutation group membership problem (AGM). McKenzie's doctoral thesis is fully devoted to those problems [1]. More recently, those problems were considered by Arvind &amp; Vijayaraghavan [3].

[1] Pierre McKenzie &amp; Stephen A. Cook. The parallel complexity of Abelian permutation group problems. 1987.

[2] Pierre McKenzie. Parallel complexity and permutation groups. 1984.

[3] V. Arvind &amp; T. C. Vijayaraghavan. Classifying Problems on Linear Congruences and Abelian Permutation Groups Using Logspace Counting Classes. 2010.