To pass a ticket in a cookie to a REST call, the cookie name must be GNPortal_gnpc_[url] where [url] is the URL of GN4.
This can be used for the authentication via cookie.
The cookie content must be a ticket transformed in base64.
Here's an example captured by Fiddler guardando il preview of an element in the User Browser. The do.ashx is called without a ticket in URL and ticket is passed (url=http://localhost/GN4 it the URL of the sample Web application).
To convert a ticket in base64 use the System.Convert ToBase64String.
To read a base64 content, use System.ConvertFromBase64String.