The client and server cannot communicate, because they do not possess a common algorithm

This was resolved. It turns out our IT Staff was correct. Both TLS 1.1 and TLS 1.2 were installed on the server. However, the issue was that our sites are running as ASP.NET 4.0 and you have > to have ASP.NET 4.5 to run TLS 1.1 or TLS 1.2. So, to resolve the issue, our IT Staff had to reenable TLS 1.0 to allow a connection with PayTrace.

So in short, the error message, "the client and server cannot communicate, because they do not possess a common algorithm", was caused because there was no SSL Protocol available on the > server to communicate with PayTrace's servers.