A call to SSPI failed, see inner exception - The Local Security Authority cannot be contacted

This means the other side is using another version of TLS and you are using an older version.
Set up security attribute to TLS12 before making the connection. This is a widely known problem, as many providers start using TLS12 (e.g. paypal,amazon and so on).

ServicePointManager.SecurityProtocol = SecurityProtocolType.Tls12;

Here is the solution, set in the registry:

[HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\SecurityProviders\SCHANNEL\KeyExchangeAlgorithms\Diffie-Hellman] "ClientMinKeyBitLength"=dword:00000200

as noted here

If you are using SslStream, then you need to explicitly set the TLS version in the AuthenticateAsClient call, for example:

ssl.AuthenticateAsClient(url, null, SslProtocols.Tls12, false);