I am a programmer and wanted to open a port for example 34000 on my public server, that my customer could use it.
What kind of security concerns that I would face?
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Generally, only "hardened" services should ever be exposed to the Internet - the service behind the port needs to be proofed against all kinds of attacks. Making an unsecure application accessible to the Internet, there's no question if you're hacked, only when.
However, limiting access to the port/service (by firewall rules or ACLs) can greatly reduce the security problem. So, if you can e.g. restrict the networks/IP ranges that can contact the service the risks may be tolerable.
You can also use a VPN-only logic to restrict access to a known group of users, regardless of where they are.
All these are security considerations and weighting the project manager has to do in accordance to the customer's requirements.