1.1 Let S = {01, 10, 11}. Note that S is a set of 2-bit strings with string 00 missing. Consider the following three One-Time Pad (OTP) variants. For each of these OTP variants state whether the resulting cipher is perfectly secure or not, and prove your answer. In other words, if your answer is yes, prove that the cipher passes Shannons perfect secrecy criterion, and if your answer is no then show that the cipher fails this criterion. In each case below the encryption and decryption procedures are as in OTP, i.e. encryption outputs a bitwise xor of the key and the message and decryption outputs a bitwise xor of the key and the ciphertext.

(a) Let M = S and K = {0, 1}^2. In other words, the message and the key are both 2-bit strings, but not every 2-bit string is a valid message.

(b) Let M = {0, 1}^2 and K = S

(c) Let M = K = S.

1.2 Are the sizes (i.e. cardinalities) of the key space and the message space in the above three cases correlated with whether or not the cipher is secure? Explain how and why.