User:Sadeghebadi

Perfect Secrecy
Intuitively, this is formulated by saying that the probability distribution of the plaintext given the ciphertext is that same as its a priori distribution. That is, Definition (perfect secrecy): An encryption scheme has perfect secrecy if for all plaintexts m ∈ P and all ciphertexts c ∈ C:                                  Pr[P = m | C = c] = Pr[P = m] An equivalent formulation of this definition is as follows.

Lemma 1 An encryption scheme has perfect secrecy if and only if for all m ∈ P and c ∈ C                                  Pr[C = c | P = m] = Pr[C = c] Proof: This lemma follows from Bayes’ theorem. That is, assume that Pr[C = c | P = m] = Pr[C = c] Then, multiply both sides of the equation by the value Pr[P = m]/Pr[C = c] and you obtain Pr[C = c | P = m].Pr[P = m]                                    ———————————————————————————————————— =  Pr[P = m]                                               Pr[C = c]  Then, by Bayes’ theorem, it follows that the left-hand-side equals: Pr[P = m | C = c] Thus, Pr[P = m | C = c] = Pr[P = m] and the scheme has perfect secrecy. In the other direction, assume that the scheme has perfect secrecy and so                                   Pr[P = m | C = c] = Pr[P = m] Then, multiplying both sides by                                   Pr[C = c]/Pr[P = m] we obtain Pr[P = m | C = c].Pr[C = c]                                    ———————————————————————————————————— =  Pr[P = m]                                               Pr[P = m]  By Bayes’ theorem, the left-hand side equals Pr[C = c | P = m] completing the lemma. Note that in the above proof, we use the fact that Pr[C = c] > 0 and the fact that Pr[P = m] > 0 Given the above, we obtain a useful property of perfect secrecy.

Lemma 2 For any perfectly-secret encryption scheme, it holds that for all m,m' ∈ P and all c ∈ C:                                  Pr[C = c | P = m] = Pr[C = c | P = m'] Proof: Applying Lemma 1 we obtain that Pr[C = c | P = m] = Pr[C = c] and likewise Pr[C = c | P = m'] = Pr[C = c] Thus, they both equal Pr[C = c] and so are equal to each other.