Proof of reserve. TransparentExchange claims that it controls at least 500,000 BTC and wants to prove this to its customers. To do this it publishes a list of addresses that have a total balance of 500,000 BTC. It then signs the statement TransparentExchange controls at least 500,000 BTC with each of the corresponding private keys, and presents these signatures as proof. What are some ways in which TransparentExchange might be able to produce such a proof even if it doesnt actually currently control 500,000 BTC? How would you modify the proof to make it harder for the exchange to cheat?