Let a M be a finite set of messages, and let S(M) denote the set of all permutations
of M (all bijective functions f : M → M). We’ll assume that if given a description of
σ ∈ S(M), both σ and σ
−1 are efficiently computable. Suppose P ⊂ S(M) is such that
∀x, y ∈ M, ∃σ ∈ P such that σ(x) = y.
(a) Show that |P| ≥ |M|. (This is easy, but makes sure you’ve parsed the definition.)
(b) Show that if |P| = |M|, then the following encryption scheme is perfectly secure,
provided you only use it once:
1
Key generation: select a random σ ∈ P;
Encryption: m 7→ σ(m)
Decryption: c 7→ σ
−1
(c)
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