I ran into this issue before with 10.2 and again with 10.2.2. For me I had to change the Authentication Settings in IIS to use 'Windows Authentication'. Worked like a charm.
Chaps,
I have had this problem but only in relation to changing the AppPool user as we have multiple WebAdaptors within multiple sites on the same IIS instance.
So traffic from one site does not affect the AppPool from another site, the AppPools required different instances/identities i.e. ArcGISWebAdaptorAppPool for the Default Site and ANotherWebAdaptorAppPool for ANother Site. The cryptography error occurs only when I moved away from the Default - ArcGISWebApatorAppPool user Identity...? So I discovered that The WebAdaptor installer set 'Load User Profile' to 'True' for the ArcGISWebAdaptorAppPool. My ANotherWebAdaptorAppPool didn't have it set up and therefore couldn't use the 'Windows Cryptographic Service Provider' therefore making Cryptographic Context unavailable as this is only available to windows account within the .NET framework. 'Load User Profile' elevates the AppPool identity to a sudo user level enabling it to use the 'Windows Cryptographic Service Provider'.