Isomorphic encryption or homomorphic encryption? Many encryption functions are said to be homomorphic: < As encryption functions are invertible, they can be considered one-to-one and onto on properly defined domains and ranges. So, my basic question is why we don't use the term "isomorphic encryption" rather than "homomorphic encryption"?

First, consider the definition of group isomorphism:

> Given two groups $(G, \otimes)$ and $(H, \odot)$, a group isomorphism from $(G, \otimes)$ to $(H, \odot)$ is a **bijective** function $f : G \to H$ such that for all $u$ and $v$ in $G$ it holds that $f(u \otimes v) = f(u) \odot f(v)$.

Now, consider a homomorphic encryption such as ElGamal cryptosystem: It takes a message from a cyclic group $G$, and outputs a pair $(c_1, c_2) \in G^2$. That is, $\mathcal{E} \colon G \to G^2$.

Notice that under this definition, $\mathcal{E}$ is **not a bijection** from the message space $G$ to the ciphertext space $G^2$. However, the decryption of ElGamal is unique.