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Key-exchange, In Particular Diffie-Hellman Key-exchange (DHKE), Is Among The Core Cryptographic Mechanisms For Ensuring Network Security. For Key-exchange Over The Internet, Both Security And Privacy Are Desired. In This Paper, We Develop A Family Of Privacy-preserving Authenticated DHKE Protocols Named Deniable Internet Key-exchange (DIKE), Both In The Traditional PKI Setting And In The Identity-based Setting. The Newly Developed DIKE Protocols Are Of Conceptual Clarity And Practical (online) Efficiency. They Provide Useful Privacy Protection To Both Protocol Participants, And Add Novelty And New Value To The IKE Standard. To The Best Of Our Knowledge, Our Protocols Are The First Provably Secure DHKE Protocols That Additionally Enjoy All The Following Privacy Protection Advantages: 1) Forward Deniability, Actually Concurrent Non-malleable Statistical Zero-knowledge, For Both Protocol Participants Simultaneously; 2) The Session Transcript And Session-key Can Be Generated Merely From DH-exponents (together With Some Public Values), Which Thus Cannot Be Traced To The Pair Of Protocol Participants; And 3) Exchanged Messages Do Not Bear Peer's Identity, And Do Not Explicitly Bear Player Role Information.