Encryption and Decryption Validation

Question

I am trying to validate encryption and decryption process. After the encryption process, I have $C$ where $$C = (z_1^x, z_2^y, m\cdot z_3^{x+y})$$

I'm also given $(z_1,z_2,z_3)$ and $(a,b)$ tuples where $$z_1^a = z_2^b = z_3$$

How to get the value of $m$, which is the original message from the given C?

I somehow need to come up with a division problem where everything except $m$ will be eliminated.

The purpose is to reverse the encryption process with decryption to get the original message $m$ from ciphertext $C$.

Answer 1

From the tuple $C$ you can compute:

• $A:=(z_1^x)^a = z_1^{ax}=(z_1^a)^x=z_3^x$.
• $B:=(z_2^y)^b = z_2^{by} = (z_2^b)^y = z_3^y$.
• $AB=z_3^x z_3^y=z_3^{x+y}$.

Now multiply the last element of $C$, by $(AB)^{-1}$ (in the field we're working in), and $m$ remains.

It's a sort of El Gamal variant.