EC: How to prevent discovery of base point after commutative operation If I have a scalar $x$ and point $B$, then I can compute $X = f(x,B)$

If the function $f$ is point multiplication, i.e. $f(x,B) = x \cdot B$, then $B$ can be determined if $X$ and $x$ are known.

Under the circumstances where $x$ and $X$ will be known, is it possible to modify the function $f$ such that $B$ cannot be determined? 

It is necessary that:


the function $f$ is commutative, i.e. $f(x, f(y, B)) = f(y, f(x, B))$
$X$ cannot be determined from $x$.
If many pairs of ($x$, $X$) are given out, it cannot be inferred that any two of the pairs were created using the same base point $B$, even if $B$ is not itself determinable.


Edit: It would be fine to limit our choice of $x$ if that would prevent $B$ from being determined.

Edit: The curve used is ed25519
If you're not tied to EC, here's one easy method: let $N$ be a composite number of unknown (secret) factorization; then:

$$f(x, B) = B^x \bmod N$$

is both uninverible, and commutative.  If $f$ needs to be a permutation (that wasn't specified), then we can select the factors $p, q$ of $N$ s.t. $p-1, q-1$ have no small odd factors (and restrict the allowable values of $x$ to small odd values $&gt;1$).

If you absolutely have to do EC, well, the obvious approach would be to actually use a pseudocurve (that is, do the EC operations in the standard way, but work on a ring rather than a field), with the ring being (yes, you guessed it) the integers modulo $N$ (where the factorization of $N$ is secret).  Yes, an operation may fail, but if the factors of $N$ are large enough, this will practically speaking never happen.  $N$ needs to be large enough so that directly factoring $N$ is infeasible (which means it's much larger than moduli we normally do EC in), however it would appear to meet your requirements.  Note that point counting on a pseudocurve doesn't work (or so we hope; if it did, then we could factor), and hence the standard way of inverting point multiplication (which involves finding the multiplicative inverse modulo the order of the curve) is unusable.