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This paper addresses the security and efficiency of key-evolving protocols. We identify forward-secrecy and backward-secrecy as the security goals for key-evolving and present two protocols to achieve these goals. The first protocol is operated in Z*p and is efficient for the secret-key holder; the second one is operated in Z*n, and is efficient for the public-key holder. For both protocols, we provide proofs and analysis for correctness, security and efficiency.