Problem 4. (20 points) The Vernam cipher (one-time-pad) works for any message space of size 2*, and has key space of size 2* as well. This works fine for sizes that are powers of 2 (a) (10 points) For any given message space m, m2,... , m, of size s (not necessarily a power of two), design a perfectly secure cryptosystem (also known as "encryption scheme") that has key space of size exactly s. Hint: think of the message space as numbers 0, 1, s-1; start with small s to try out some examples (b) (10 points) Prove that your cryptosystem is perfectly one-time secret, in the same sense as Definition 2.6 in Rosulek's text http://web.engr.oregonstate.edu/-rosulekm/crypto/ . Note: your proof will follow the same steps as the proof of Theorem 2.8. Please do NOT write down those steps in your solution. The only difference will be in Claim 2.3. So all we want you to do is write down the analogue of Claim 2.3 for your cryptosystem and prove it. It should look something like this my-system-real my-system-random code goes here return something code goes here return something