"No input file specified" usually shows up when you're running PHP-FPM and the whatever.php file your server is sending you to doesn't exist, and therefore PHP-FPM can't execute the file ... because none is available.
It's somewhat akin to PHP/Apache triggering a generic "500 Internal Error".

Typical causes:
a) the server's configuration for PHP+apache/nginx has been mangled, such as by the server administrator doing a PHP upgrade incorrectly
b) you've changed the paths in your /includes/configure.php file to point to invalid directories
c) you've accidentally renamed or drag-and-dropped folders in FTP so that they're in wrong places ... or deleted them altogether
d) you've set permissions on some files or directories that are either too permissive or too restrictive, and the server's security system is denying PHP access to those files thus triggering the error

Or, given recent hype about HTTPoxy security alarms, perhaps your hosting company has incorrectly applied a patch to the server config?

The server's PHP error logs quite probably contain the actual real details of the problem.