In this paper, a new efficient public key multi-messages multi-recipients signcryption (PK-MM-MRS) scheme for provable secure communications has been proposed. The proposed scheme signcrypted multiple messages for each recipient using a multi-key generator. The multi-key generator played an important role for the communication overhead and computational cost saving in the proposed PK-MM-MRS scheme. This scheme has an efficient communication overhead and computational cost saving compared to Zheng multi-recipient signcryption scheme. The numerical values for the saving in communication overhead using the proposed scheme are 38.24% for two messages, two recipients and 67.97% for three messages ,three recipients than does Zheng multi-recipient signcryption scheme with same parameters. The proposed scheme costs 50% and 75% less in computation cost for two and four signcrypted messages over Zheng multi-recipient signcryption scheme. The saving in communication overhead and computational cost is increased as the number of messages and recipients increases.