Consider the following cryptosystem: K = {A, B} Pr(A) = 2/3 Pr(B) = 1/3 P = {0, 1} Pr(0) = 3/5 Pr(1)

Consider the following cryptosystem: K = {A, B} Pr(A) = 2/3 Pr(B) = 1/3 P = {0, 1} Pr(0) = 3/5 Pr(1) = 2/5 C = {a, b} EA(0) = a EA(1) = b EB(0) = b EB(1) = a a) Compute Pr(a) and Pr(0|a). Solution (a) Pr(a) = Pr(0) · Pr(A) + Pr(1) · Pr(B) = 8 15 . Use the Bayes’ theorem to compute, Pr(0|a) = Pr(0) · Pr(a|0) Pr(a) = 3 5 · 2 3 8 15 = 3 4 . b) Is this system a perfect cryptosystem ? If not, what probabilities you would change to make it perfect ? Solution This is not a cryptosystem with perfect secrecy. We need to change the key distribution. Due to Shannon for a cryptosystem with |K| = |C| = |P| we must have that P r(A) = P r(B) = 1/2. In this case P r(a) = 1/2(P r(0) + P r(1)) = 1/2 = P r(b