Access to http restricted area using private key

Question

In my Apache configuration (Ubuntu Server), i have the following settings:

    AddExternalAuth pwauth /usr/sbin/pwauth
    SetExternalAuthMethod pwauth pipe

    <Directory /secure>
        Options Indexes FollowSymLinks Includes ExecCGI
        AllowOverride All
        AuthType Basic
        AuthName "Restricted Content"
        SSLRequireSSL
        AuthBasicProvider external
        AuthExternal pwauth
        Require valid-user
    </Directory>

And when I browse to http://localhost/secure, I can access using user&pass of my Ubuntu users.

Now I want to access using username & private key of my users (of course not through browsers). Is it possible?

I already tried the following configuration:

    AddExternalAuth sshd /usr/sbin/sshd
    SetExternalAuthMethod sshd pipe

    <Directory /secure>
        Options Indexes FollowSymLinks Includes ExecCGI
        AllowOverride All
        AuthType Basic
        AuthName "Restricted Content"
        SSLRequireSSL
        AuthBasicProvider external
        AuthExternal sshd
        Require valid-user
    </Directory>

But with no success.

Is there anything I can do, or HTTP authentication does not support private key?