Throughout, I'm only interested in structures with domain $\mathbb{N}$, no primitive relations, and at least $0,\mathsf{Succ}$ as primitive functions. The length of $m\in\mathbb{N}$ is $\lfloor 1+\log_2m\rfloor$, and the length of a term is the number of symbols occurring in it.
Suppose $\mathfrak{N}=(\mathbb{N};...)$ is a structure of the type above. Given a set $X\subseteq\mathbb{N}$, say that the $\mathfrak{N}$-anonymity of $X$ is the (growth rate of the) function $A^\mathfrak{N}_X$ sending each $k$ to the smallest $n$ such that for every $a,b<2^k$ with $a\in X$ and $b\not\in X$ there are terms $s(x),t(x)$ in the language of $\mathfrak{N}$ with length $<n$ such that $s(a)=t(a)$ but $s(b)\not=t(b)$ or vice versa. For every $\mathfrak{N}$ and $X$, the $\mathfrak{N}$-anonymity of $X$ is at most exponential, so this is an extremely coarse notion. On the other hand, it's not entirely trivial; for instance, unless I made a silly mistake, it yields a natural way of showing that we cannot determine parity using addition alone in polynomial time.
My question is:
What is this notion actually called, and what is a good source on it?
(I'm asking here as opposed to cst.se since I suspect my interests in this topic have less to do with computer science than with mathematics. In particular, while not at first I'm eventually going to be interested in extensions of this notion to structures other than versions of $\mathbb{N}$.)
