3. Equivalent definition of Perfect Secrecy (10 points). In the lecture we defined the perfect security for any private-key encryption scheme (Gen, Enc, Dec) as follows. For any message m, cipher-text c, and a priori probability distribution M over the set of messages, we have: P[M=m|C=c] =P[M=m] Show that the above definition is equivalent to the following alternative definition. For all messages m, m', cipher-text c, and a priori probability distribution M over the set of messages, we have: P[C=c|M=m] =P [C=c|M=m'], Remarks: (1) Proving equivalence means that you have to show that the first definition implies the second definition. And, the second definition also implies the first definition. (2) Additionally, in this problem, for simplicity, assume that in the the probability expressions no "division by error occurs. 3. Equivalent definition of Perfect Secrecy (10 points). In the lecture we defined the perfect security for any private-key encryption scheme (Gen, Enc, Dec) as follows. For any message m, cipher-text c, and a priori probability distribution M over the set of messages, we have: P[M=m|C=c] =P[M=m] Show that the above definition is equivalent to the following alternative definition. For all messages m, m', cipher-text c, and a priori probability distribution M over the set of messages, we have: P[C=c|M=m] =P [C=c|M=m'], Remarks: (1) Proving equivalence means that you have to show that the first definition implies the second definition. And, the second definition also implies the first definition. (2) Additionally, in this problem, for simplicity, assume that in the the probability expressions no "division by error occurs