If my key size is as large as the data I'm encoding, is it trivial to devise a theoretically secure homomorphic encryption scheme for integers (or else any finite/infinite group with order) that supports $+$ as an operation on encrypted data? Plus points if your scheme is computationally inexpensive or easy to understand.
Formally, $e(k,a+b)$ can be computed quickly from $e(k,a)$ and $e(k,b)$ where $k$ is the key. Theoretically secure means it should be impossible to determine any $a_i$, given a large number of samples $e(k,a_i)$ unless $k$ is known. Also $e(k,a)$ shouldn't be constructible for known $a$ unless you know $k$
I tried coming up with an example, which I couldn't but it does appear quite doable.
A use case for this could be say secure online voting, wherein the vote count is a small data value but it would be good if the votes can be added in by servers who themselves don't have access to the vote count.