We exploit the relationships between signature schemes and key agreement protocols; and propose a high performance identity-based (ID-based) key agreement protocol based on strong pairing challenge-response signatures. The latter is the first of its kind in ID-based cryptography and is of interest in itself. Using the proof technique of signature unforgeability against adaptive chosen-message attack, our protocol fully supports Session-Key Reveal queries and partially supports Session-State Reveal queries (which leaks ephemeral secret and keying material for session key derivation), without gap assumption or any unrealistic restriction. We show how to incorporate KGC forward secrecy so the past session keys are not compromised even the adversary gets the master secret key of the Key Generation Center (and the private keys of all users). Both proposals are efficient and have the strongest security among other unbroken identity-based two-party two-message protocols. Inspired by ring signatures and motivated by the need for a better anonymous roaming mechanism, we extend our basic protocol to support key agreement among spontaneous anonymous groups (SAG). To the best of our knowledge, this is the first ID-based SAG key agreement protocol with bilateral privacy.