Reversible Merge of Integer Hash Values

Question

Context:

I am working with a tree-like data structure. I would like every node in the tree to have an integer hash value that is the result of combining the integer hash values for the node's children. Additionally, I'd like to be able to replace one of the node's children in constant time (i.e., not doing a full recombination of hash values) as the branching factor for the tree could be quite large.

Question:

I am looking for a set of functions ($combine$, $separate$) that operate on fixed width integers and have the following properties.

Let $X$ be some set of integers with fixed width $b$.

(1) $combine(X)$ outputs an integer with width $b$

(2) $combine(X)$ operates in $O(|X|)$ time complexity

(3) $$combine\Bigl(\left\{t, separate\bigl(combine(X), x\bigr)\right\}\Bigr) = combine\Bigl(\bigl(X \setminus \{x\}\bigr) \cup \{t\}\Bigr)$$ where $x \in X$ and $t$ is some integer of width $b$.

(4) $separate(X, x)$ operates in $O(1)$ time complexity

(5) The outputs of $combine$ are going to be used as hash values so an algorithm with a "good" distribution of output values to avoid collisions is ideal. That said, I'd define good as anything better than trivial addition and subtraction (see below).

Initial thoughts:

A trivial implementation would have $combine$ and $separate$ be addition and subtraction (with some modulo logic); however, this obviously won't have a good distribution for tightly bunched input values.