It appears that in some fairly recent version of OpenSSH, the support for the
~/.ssh/authorized_keys2 file was removed (along with the
known_hosts2 file). It had apparently been deprecated in preference to the
~/.ssh/authorized_keys file a while ago. This caused me some grief when my ISP silently upgraded OpenSSH recently and my automatic backup scripts (which rely on key authentication) stopped working. Renaming the file fixed the problem.