Question

# 1. Consider each of the following encryption scheme and state whether the scheme provides perfect secrecy...

1. Consider each of the following encryption scheme and state whether the scheme provides perfect secrecy or not. Justify your answer by giving a detailed proof if your answer is Yes, a counterexample if your answer is No.An encryption scheme whose plaintext space consists of the integers ?={0, 1, ..., 12}and key generation algorithm chooses a uniform key from the key space . Suppose mod 13 and mod 13?={0, 1, ..., 12}????(?)=?+?????(?)=?―?

Enck(m) = m + k Deck(c) = c -k satisfies perfect secrecy

Proof: For any m, m0 ,c ∈ G, we have:

Pr [EncK(m) = c]

= Pr [K + m = c]

= Pr [K = c - m]

Since c-m is a constant value for chosen m and c, we are really considering the probabiltiy that a randomly chosen key is equal to their distance. Since all keys are equally likely and chosen from the key space G, this probabiltiy is equal to:

Since the choice of m was arbitrary, we have that Pr [EncK(m) = c] = Pr [EncK(m0 ) = c],

which is one of the definitions of perfect secrecy.

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