I'm in need of some distributed processing, the cheapest way for me to get that processing power is renting some dedicated servers from a company with specialized hardware for this.

The servers that they have aren't internally connected in any way, they might not even be on the same continent as each other. All they have is an internet connection and a subdomain address (like node523.bunchofservers.com). Well I could also get dedicated IP's but that would cost extra.

Anyway, one of them would be the coordinator of all this processing and would have to be barking orders at other nodes, the data transferred is not sensitive information and doesn't need encryption.

Would it be safe for me to just hardcode the coordinator servers subdomain address into the processing nodes and just accept whatever orders come from that address without any other form of authentication?

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No. Network connections can be spoofed in a variety of ways. The impact of this is not something that we can say with the information provided; you'll have to evaluate that; but spoofing is technically possible. If that would be undesirable, use a suitable form of authenticated network connections, such as TLS with public-key authentication.

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    Though with control of both ends you can pre-agree a secret between the nodes and not need public-key. Though most implementations is geared towards certificate based communication anyway. – ratchet freak Sep 3 '18 at 8:21

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