I just want to point out to a problem I have just resolved. Customer reported a problem when accessing Sharepoint 2010 site that is published using Forefront TMG 2010. Sharepoint listener is configured to use forms-based authentication with Persistent Cookies enabled for both public and private computers. For those who wants to know how this works you can read my article here.
The problem was that when the Private computer was selected in the authentication form when accessing site it worked fine but when the Public or Shared computer was selected the user could not be authenticated. The TMG log showed the following error:
12302 - The server denied the specified Uniform Resource Locator (URL). Contact the server administrator.
When the listener was modified to use Persistent Cookies only for the Private computers then the authentication worked also for Public computers but then the desired authentication cookie was not saved to the local client cache. The problem appeared on IE, Firefox and on different client computers so it was not related to the local client cache or something like that. TMG and Sharepoint servers were restarted but it also did not help.
The problem was resolved only by recreating the Sharepoint listener on TMG using all the same properties as before. Now both the Public and Private computers work with Persistent Cookies enabled.