I had the same problem when attempting to update my iPad Air 2 using iTunes (12.7.0.166) on Windows 10 (tried uninstalling/reinstalling iTunes, considered downgrading iTunes, saved a new hosts file as directed by Apple Support pages, even used different USB cables in case the issue was from using an aftermarket cable...) and the ONLY way I was able to get my iPad out a perpetual state of the 3503 error was, sadly, to use a Mac. The software update and restore to factory settings (only option by this point) all went fine, and after I was able to restore from a backup saved on my PC earlier in the day.
The biggest difference I noticed was that the Mac actually DOWNLOADED the 2.34 GB Software Update (which took almost 20 minutes), whereas on my PC, iTunes was proceeding straight to "Extracting Software" without any obvious download to my hard drive happening.
Side note, I did not try doing a WiFi update because I did not have the necessary disc space on my iPad, but I suspect (as someone else has mentioned) that it would have worked fine as it would bypass the "Microsoft" factor.
Hopefully Apple fixes this soon, it already sounds like it is popping up for a lot of people.