We study a secure searchable public-key encryption scheme with a designated tester (dPEKS). The contributions of this paper are threefold. First, we enhance the existing security model to incorporate the realistic abilities of dPEKS attackers. Second, we introduce the concept of "trapdoor indistinguishability" and show that trapdoor indistinguishability is a sufficient condition for thwarting keyword-guessing attacks. This answers the open problem of how to construct PEKS (dPEKS) schemes that are provably secure against keyword-guessing attacks. Finally, we propose a dPEKS scheme that is secure in the enhanced security model. The scheme is the first dPEKS scheme that is secure against keyword-guessing attacks.