In IEEE Communication Letters, Kim et al. and Sun et al. proposed several e-mail protocols and claimed that they provide perfect forward secrecy. In this paper, we show that under the assumption that the e-mail server is also malicious, Kim et al.'s two e-mail protocols do not provide perfect forward secrecy. While one of Sun et al.'s e-mail protocol provides perfect forward secrecy, it does not provide message authentication. We thus propose an improvement to Sun et al.'s e-mail protocol so that it provides both perfect forward secrecy and message authentication.