To address security and privacy issues in messaging services, we present a public key signcryption scheme with designated equality test on ciphertexts (PKS-DET) in this paper. The scheme enables a sender to simultaneously encrypt and sign (signcrypt) messages, and to designate a tester to perform equality test on ciphertexts, i.e., to determine whether two ciphertexts signcrypt the same underlying plaintext message. We introduce the PKS-DET framework, present a concrete construction and formally prove its security against three types of adversaries, representing two security requirements on message confidentiality against outsiders and the designated tester, respectively, and a requirement on message unforgeability against the designated tester. We also present three extensions, analyze the efficiency of our PKS-DET construction and extensions, and compare them with related schemes in terms of ciphertext sizes and computation costs of signcryption (encryption), unsigncryption (decryption) and ciphertext equality testing. Experimental results further confirmed the practicality of our construction.