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In 2003, Boyd et al. have proposed two deniable authenticated key establishment protocols for Internet key exchange (IKE). However, both schemes have been broken by Chou et al. in 2005 due to their susceptibility to key-compromise impersonation (KCI) attack. In this paper, we put forward the improved variants of both Boyd et al.'s schemes in order to defeat the KCI attack. On top of justifying our improvements, we further present a detailed security analysis to ensure that the desired security attributes: deniability and authenticity remain preserved.