Everyone complaining about this needs to understand that if you win Powerball or Mega Millions in the following states you can legally remain completely anonymous:
Delaware, Kansas, Maryland, North Dakota, Ohio, South Carolina, and Texas. (Georgia just passed a lottery anonymity law in their senate and this Powerball ruling may have just changed New Hampshire.)
There are many more states that allow a trust to claim a lottery prize, which can also keep the winner's name anonymous.
Therefore, you cannot claim that winner's names must be published to maintain the integrity of the lottery, because it's simply not supported by evidence. Auditors will always have access to the winners' information, it is their job to make sure the lottery is legitimate. Making the name public has nothing to do with the integrity of the lottery, they just want the media spectacle for free advertising so they can sell more tickets.
EDIT: As long as there are states that allow winners to remain completely anonymous, you have a good argument that your state lottery has no valid reason to publish your name other than advertising.
EDIT2: Updated state list to be more accurate.