Strange questions about SSH remote Logon

For details about SSH remote logon, refer to Linux Enterprise Application-Linux server application. If you want to log on to the Server (Ubuntu Server 9.04) by using SSH in your laptop (Ubuntu Desktop 9.04), you can use password authentication to log on to the Server. However, considering that the Server is on the internet, you are worried about security issues, so I want to use the dsa authentication method? Why? Silly? The ub key is added to the server's ~ /. Ssh/authorized_keys, and then modify AuthorizedKeysFile % h/of/etc/ssh/sshd_config /. ssh/authorized_keys and PasswordAuthentication no. Then, we found a strange phenomenon. The server prompts a public key error during ssh remote logon without a user logon, remote ssh can be used to log on when a user logs on to the server. After PasswordAuthentication is changed to no, remote ssh can be used to log on to the server.
How can I access the server through remote ssh regardless of whether the server is logged on?