Hi guys, I'm having exactly the same problem as described by everyone before excepts I get error 1669 instead of error 3.
My iPhone 5S started to say "Searching..." in the morning after charging it all night. I removed the SIM card and it was still saying "Searching...". It didn't realize that there was no SIM inside. I reset network settings, reboot the phone, etc and but it was still on "Searching". So, I decided to do a restore... It's the worst thing I did. It stucks at 66% (aprox) in the iPhone progress bar while restoring and iTunes fails with error 1669. I've never seen error 3. But I have exactly the same symptoms as you described in the 30 pages of this thread.
I followed the "freezer" tip with no luck. I always get the same error. It's weird that error 3 and error 1669 seem to be the same. I read the logs you posted here and compared with my iPhoneUpdate.log's and they seems to be very similar.
I bought the iPhone on October 2013 (it's on warranty but there's no support for iPhone in my country). I updated iOS always via OTA. After 7.1 update I started to have some battery drain problems, so I decided to do a DFU restore to see if this could solve the problem. It was the only update I did to the phone using iTunes. It was successfull. 10 days later I started to experience the problem described above (Searching and 1669 when restoring).
If it helps, I'm posting 2 logs. The first one is when I did a successful restore. The second is one of the many attempts to restore without luck.
_______________
Succesful DFU restore (12 days ago):
https://www.dropbox.com/s/c1u38htj8nthzob/iPhoneUpdater%208.log
_______________
Unsuccessful DFU restore (2 days ago to now):
https://www.dropbox.com/s/hb5jyieb1rstazx/iPhoneUpdater%205.log
Any suggestion, idea, or whatever will be very appreciated!
Thanks!