Part 1: Using PGP or similar encryption schemes, we can encrypt a message so that the same message can be opened by two or more recipients, each using his or her own private key, without materially increasing the size of the encrypted package. Carefully explain how that can be accomplished. (Hint: the actual message is encrypted using a symmetric key.)

Part 2: We would like to protect biometric data used for authentication because, once compromised, it is compromised forever. Explain why one must use reversible encryption, and not a cryptographic hash, to store biometric information. (Adapted from Computer Security by Goodrich and Tamassia.)

Part 3: A suitcase has a combination lock with four wheels. Each wheel has ten digits, 0 through 9 How many different combinations are there? Assume that a thief can try one combination every second, and that the thief will get lucky after trying about half the combinations, how long will it take the thief to open the briefcase?

If each wheel of the briefcase lock has only eight digits: 0, 1, 2, 3, 4, 5, 6, 7; how many different combinations are there? How long will it take our thief to open the lock?

Part 4: Suppose that a lock like the one described above has only two numbers, 0 and 1, on the wheels, and there are four wheels. How many combinations are there? What happens if an additional wheel is added? How many combinations, and how long to "crack?"

In general, what does adding another wheel do to the number of combinations?

Part 5: Explain what, if anything, the previous two exercises have to do with encryption and what, if anything, you learned from them.

Something to Try: You do not have to do this, but you will find it instructive. Suppose you have a computer that can make 10⁵⁰ guesses per second against against a 256-bit binary combination like the lock in Part 6. About how many years will it take to try half the combinations? You may express the answer as a power of ten. Google is your friend when you need to do arithmetic like this. Just type an expression into Google and you'll get an answer. Exponentiation is represented by ^ so 10⁵⁰ is 10^50.

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