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Nowadays there are many social networking services supporting three-party communication such as Skype, Line, and Facebook Messenger. To ensure the message security, a cryptographic encryption scheme is a commonly adopted measure. However, the traditional asymmetric encryption only allows one designated recipient to decrypt the ciphertext with his/her private key. It is thus difficult for two parties to share the same ciphertext without exposing their private keys. In this paper, the authordoi:10.3390/app9071452 fatcat:2vmgkewcobc7xlnjivoft6qvmy