We report a fatal flaw of CSP-ElG scheme, one of public key encryptions based on conjugacy search problem proposed in INSCRYPT 2010. It does not satisfy the security property claimed as it is. We also discuss imperfections of security proofs of the other proposals: CSP-hElG and CSP-CS schemes. Following the technique given by Gennaro et al. to smooth a distribution of DH transform outputs, we introduce a computational assumption related to monoid actions and fix the CSP-ElG scheme using a universal hash function and the leftover hash lemma.