How to Update Non-Membership Witness upon deletion in RSA Dynamic Universal Accumulator The Universal Accumulator paper mentions the following for updating non-membership witness upon member deletion.

Choose $r$ such that $a\hat x − rx ∈ Z$$2^l$

What is Z$2^l$ and how can we efficiently evaluate $r$?
I suppose we need to brute force search by guessing values of r, is there any room for optimization here? Perhaps by making an educated guess about the value of r.
Update:
Can it be that $a\hat x − rx$ can be construed as $a\hat x$ $mod x$? If yes, then we can find
$\hat a = a\hat x$ $modx$
And,
$r = (a\hat x - \hat a)/x$
Am I right in assuming this?
The updated Non-Membership Witness ($\hat a, \hat d$) upon deleting a member $\hat x$ can be calculated as follows:
$\hat a = a\hat x\mod x$
$r = (a\hat x - \hat a)/x$
$ \hat d = (d\hat c$ -r )$\mod n$
Hope this is useful to future developers trying to deconstruct these equations.