The irs is kind of in charge of some taxes. An affidavit is a form you bring to get notarized, usually at your local bank. getting that form notarized requires your social security number, but it proves that you are the person they mean to give the prize too, and I assume the irs gives them a tax return for something the lines of charitable giving for giving you the phone.
It sounds really sketchy giving your social security #, but that's how it works. Don't do it if you feel AT ALL that it is sketchy. I was weirded out by the process to get my gear, but I got it and my identity wasn't stolen.