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With the emergence of diverse group-based services, multiple multicast groups are likely to coexist in a single network, and users may subscribe to multiple groups simultaneously. However, the existing group key management (GKM) schemes, aiming to secure communication within a single group, are not suitable in multiple multicast group environments because of inefficient use of keys, and much larger rekeying overheads. In this paper, we propose a new GKM scheme for multiple multicast groups, called the master-key-encryption-based multiple group key management (MKE-MGKM) scheme. The MKE-MGKM scheme exploits asymmetric keys, i.e., a master key and multiple slave keys, which are generated from the proposed master key encryption (MKE) algorithm and is used for efficient distribution of the group key. It alleviates the rekeying overhead by using the asymmetry of the master and slave keys, i.e., even if one of the slave keys is updated, the remaining ones can still be unchanged by modifying only the master key. Through numerical analysis and simulations, it is shown that the MKE-MGKM scheme can reduce the storage overhead of a key distribution center (KDC) by 75 percent and the storage overhead of a user by up to 85 percent, and 60 percent of the communication overhead at most, compared to the existing schemes.