Zero-Knowledge Challenge-Responce Protocol
Reading the original paper, I figure out the question.
This voting scheme employed the well-known undeniable signature scheme, proposed by Chaum and Van Antwerpen in 1989 (or Chaum 1990 or Chaum and Van Antwerpen 1991).


KeyGen: The RA is a signer and has a public key $X = g^x$ and a secret key $x$
Sign: For a message $m \in \mathbb{G} = \mathbb{Z}_p$, the signature is $\sigma = m^x \in \mathbb{G}$. (In your case, $m = r_{id}$.)
Cofirmation: The verifier (the polling station) and the signer (the RA) interacts to verify a token $(r_{id},\sigma)$ from a voter is correct or not.


In confirmation, the RA proves that $\log_g(X) = \log_{r_{id}}(\sigma)$. You can find the standard $\Sigma$ protocol for this confirmation, say, the $\Sigma$ protocol for the DDH language.

You can find the $\Sigma$ protocol for the DDH language in the lecture notes available at the web.


Section 5 of Berry Schoenmakers: Lecture Notes - Cryptographic Protocols
Lindell: Sigma protocols and Zero Knowledge (Session 5 of Winter School on Secure Computation and Efficiency)