Can encbup be used for automated backups? Seems like it's using symmetric encryption, so that the passphrase would need to be entered or passed as an argument on each run.
Not sure if it meets all your requirements, but duplicity [1] has support for asymmetric encryption via gpg pub/priv key pairs.
[1] http://duplicity.nongnu.org/
EDIT: just noticed the article mentioned duplicity as insufficient.