Hello!
I was tried more than 10 times to restore, in recovery mode and DFU mode but no success!
My phone was not under warranty and i have nothing to lose...
I opened it, cleaned the motherboard with isopropyl and heating with hairdryer for 5-6 minutes, especially over the baseband (near the sim card slot).
After that I tried to restore, and its working!!!
Sometimes it has problems with memory (out of memory alert) but after one month still working properly.
If your iphone 5s isn't under warranty, try this method, maybe will work....!!!