We explore ways to optimize online, permutation-based authenticated encryption (AE) schemes for lightweight applications. The lightweight applications demand that AE schemes operate in resource-constrained environments, which raise two issues: 1) implementation costs must be low, and 2) ensuring proper use of a nonce is difficult due to its small size and lack of randomness. Regarding the implementation costs, recently it has been recognized that permutation-based (rather than block-cipher-based) schemes frequently show advantages. However, regarding the security under nonce misuse, the standard permutation-based duplex construction cannot ensure confidentiality. There exists one permutation-based scheme named APE which offers certain robustness against nonce misuse. Unfortunately, the APE construction has several drawbacks such as ciphertext expansion and bidirectional permutation circuits. The ciphertext expansion would require more bandwidth, and the bidirectional circuits would require a larger hardware footprint. In this paper, we propose new constructions of online permutation-based AE that require less bandwidth, a smaller hardware footprint and lower computational costs. We provide security proofs for the new constructions, demonstrating that they are as secure as the APE construction.