Mathematical formalisation of encryption and decryption

Question 1

Mathematical formalisation

Answer 1

1. A - finite set of alphabets
2. M ⊆ A∗ is the message space. m ∈ M is a plaintext (message).
3. C is the ciphertext space, whose alphabet may differ from M
4. K denotes the key space of keys
5. Each e ∈ K determines a bijective function from M to C, denoted by Ee. Ee is the encryption function/
   transformation. You can write we will write Ee(P) = C or, equivalently, E(e, P) = C.
6. For each d ∈ K, Dd denotes a bijection from C to M.
   Dd is the decryption function.
7. Ee is the encryption function and Dd is the decryption function.

Question 2

Encryption Scheme

Answer 2

1. An encryption scheme (or cipher) consists of a set {Ee | e ∈ K} and a corresponding set {Dd | d ∈ K} with the property that for each e ∈ K there is a unique d ∈ K such that Dd = E−1e
2. Dd (Ee(m)) = m for all m ∈ M.
3. The keys e and d above form a key pair, sometimes denoted by (e, d). They can be identical (i.e., the symmetric key).
4. To construct an encryption scheme requires fixing a message space M, a ciphertext space C, and a key space K, as well as encryption transformations {Ee | e ∈ K} and corresponding decryption transformations {Dd | d ∈ K}.

Question 3

Bijection example lecture 2.2 (slide 7)

Answer 3

M = {m1, m2, m3} and C = {c1, c2, c3}
there are 3! = 6 bijections from M to C
The key space specifies the number of transformations - in this example there were 6 keys so, check all the possible transformations from m -> c when E1, E2, E3, E4, E5 and E6 are used.