How to protect private blockchain with public blockchain

Question

Private blockchain has the risk of 51% attack, while public blockchain is pretty safe. So can we protect private chain with public chain? I got an idea, that we can save block height and block hash, every 100 blocks. We save 1,101,201... block height and hash on public chain, and we verify it. But it still has risk, before 101 block is protected, 2-99 blocks still have risk. Is there a better way to solve it?