(30 points) For each of the following symmetric-key encryption schemes, state the strongest of the following security definitions that it necessarily satisfies: (2) chosen-plaintext (CPA) security, (1) EAV security, or (0) neither of these. In all cases, the key-generation algorithm Gen (1n) outputs a uniformly random key k{0,1}n. Give some sound intuition why the scheme satisfies the definition you identified, though you do not need to give a formal proof. If applicable, also show that the scheme does not necessarily satisfy the next-strongest definition, by giving an efficient adversary that has non-negligible advantage in the corresponding game. Also, recall that (Init, NextBit) is a secure stream cipher, Init takes in an initialization vector, and outputs an initial state; st, and NextBit(sti)=(sti+1,y); i.e. it outputs an updated state and a bit y. (a) Enck(m{0,1}) lets st0=Init(k) and generates a string p{0,1}m by iterating NextBit starting from st0, where (Init, NextBit) is a secure stream cipher. It outputs ciphertext c=mp. (b) Enck(m{0,1}3n) chooses random r{0,1}n and outputs ciphertext c=(r,(G(r)k)m), where G is a PRG with stretch (n)=2n. 2 (c) Enck(m{0,1}n) lets pk=G(k) for p,k{0,1}n where G is a PRG with stretch (n)=2n, outputs ciphertext c=mp, and replaces k with k for the next call (so Enc is stateful)