Your question does not explicitly indicate whether you are limited to using ADFS to implement single sign-on. ADFS is definitely one way to do this. You can look at OpenID-LDAP (it was at www.openid-ldap.org, but the project is no longer working) and other identity providers as alternatives for implementing single sign-on.
One option is to implement an OpenID provider that uses integrated Windows authentication. They can install it in the DMZ, exposing it to the Internet, not ADFS. It can enable single sign-on in Internet Explorer and Chrome.
The implementation of the OpenID provider is a non-trivial matter. You will need to pay close attention to safety. Fortunately, there are a number of infrastructures, such as DotNetOpenAuth , that can help you.
When using OpenID, your "cloud" application will act as an OpenID link and get the declared identifier among other attributes from the OpenID provider. You must store this in your database to uniquely identify the user. You can choose an OpenID provider so that it can also provide your cloud application with minimal information such as email address, person’s name, etc.
The advantage of using OpenID is that your cloud application can continue to support other well-known OpenID providers, such as Google and Yahoo, without much change.
In the end, you will need to ask your customers which technologies are convenient for them. You will find that the lack of trust (a business trait) between organizations is most often a problem, not a technology.
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