So I was reading a paper on homomorphic encryption, and it in turn introduces some concepts that I didn't know much about before (primarily groups).
I have a few questions but I'll first post the pertinent paragraph in the paper:
"Earlier, homomorphic encryption was defined as a form of encryption where a specific algebraic operation performed on plaintext is equivalent to another algebraic operation performed on its ciphertext. In mathematics, these operations are called mappings or functions and are operation preserving (OP) mappings. For example, if (G,o) and (H,$^*$) represent groups, an OP mapping h:(G,o) $\to \!\,$ (H,$^*$) is called a homomorphism from (G,o) to (H,$^*$). The groups (G,o) and (H,$^*$) represent sets of data and the mapping (function) that maps the set (G,o) onto or into the set (H,$^*$) is h. This function is an operation that preserves the structure from one set of data to the other set of data.
....
To further explain how homomorphism works, we consider two sets of data. One is the set of positive real numbers, R$^+$, and the other set is the logarithms of this set of real numbers. On these sets, the multiplication of real numbers and the addition of logarithms and homomorphic operations. If we consider any real positive numbers x, y, z, if x × y = z, then log(x) + log(y) = log(z). "
So my questions are as follows:
Using the notation from the first paragraph, would it be mathematically correct to write the following:
The operational preserving mapping h: (R$^+$, ×) $\to \!\,$ (L, +) (where I define L to be the set of logarithms of the positive real numbers) is called a homomorphism from (R$^+$, ×) to (L, +) (assuming (L,+) is a group).
Furthermore, if the function h represented an encryption function (that I'll call e), and M my set of plaintext messages and C my set of ciphertexts, could I say the following?
My encryption function will be called e
e: (M,o) $\to \!\,$ (C,*) is a group homomorphism from (M,o) to (C,$^*$), and would the reverse automatically be true?
Meaning is
d: (C,*) $\to \!\,$ (M,o) a group homomorphism from (M,o) to (C,$^*$), where d is my decryption function and o and $^*$ are operations on the sets M and C respectively?