I am having the same problem. I tried updating from iTunes 12.6 to the latest 12.7 and still have the same problem.
My iPhone is a 6S Plus running IOS 10.3.1.
I have never done anything unusual to this phone. No jailbreak or anything else. I've rebooted, reinstalled iTunes, and always get the 3503 error. I turned off my antivirus software and anything else that was running. My HOSTS file has no entries in it.
For all the money i have paid Apple over the years I expect a lot less hassle. in fact, that is the reason I choose Apple, because it is supposed to be easy and stable. I use my phone for banking, phone, messages, browser, weather, and that's about it. This should be easy.
Of course I don't expect Apple to give a **** or reply here, or make any effort to assist me. This is purely a way for me to vent. Samsung Galaxy Note is looking better to me each day.