The problem is really much worse than this. You don't merely need a non-repeating nonce (the way you can get away with a GCM nonce that increments by 1 every session): you need an unbiased nonce.
I believe an natural segue here is to remind people about cryptopals (especially set 8). Ie., I don't have the chops and wouldn't attempt to writeup EC/DSA nonce bias and partial key exposure attacks better than you all--not to mention the challenges regarding GCM. Cheers.