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These errors typically occur when security software interferes with the restore and update process. Follow Troubleshooting security software issues to resolve this issue. In rare cases, these errors may be a hardware issue. If the errors persist on another computer, the device may need service.
Also, check your hosts file to verify that it's not blocking iTunes from communicating with the update server. See iTunes: Advanced iTunes Store troubleshooting—follow steps under the heading Blocked by configuration (Mac OS X / Windows) > Rebuild network information > Mac OS X > The hosts file may also be blocking the iTunes Store. If you have software used to perform unauthorized modifications to the iOS device, uninstall this software prior to editing the hosts file to prevent that software from automatically modifying the hosts file again on restart.