I'm working on an internet election system that requires the shuffling of ballots accompanied by an interactive proof of the legitimacy of the shuffle. I am working on this paper and I am stuck at the part outlined below:
By releasing the single value $(r'-r'')\mod(p-1)$, the two ElGamal pairs $(x',y')$ and $(x'',y'')$ can be shown to have the same decryptions without any linkage or association to the original ElGamal pair $(x,y)$.
I managed to get the value $(r'-r'')\mod(p-1)$ outlined above but I am not sure how to use this value to prove that both re-encryptions have the same decryption.
Thank you for the time 😊,