This is actually impossible without hardware encryption. Apple should have no problem with this since they have control over their hardware, but they don't do anything to prevent fake scores from showing up in their leaderboards. Xbox is a great example of being able to trust the client. The console tells the server that it has unlocked an achievement by using a signed request which was signed by the secure chip. The same chip verifies that the games you are playing are legitimate, etc. It's a very important piece of hardware and it would likely be the Xbox's demise if that secure chip suddenly became insecure.

Running the game on the server is the only option as far as I know for securing the integrity of the data. You could add a private key to your app, but then you're shipping the private key to the client.