The Public Key Cryptography

Abstract

A cryptographic system consists of a one to one enciphering transformation f from a set $\mathcal{P}$ of plaintext message units to a set $\mathcal{C}$ of ciphertext message units. For example, for a fixed N- letter alphabet, identified with ℤ/Nℤ, consider the mapping f : $\mathcal{P}$ = ℤ/Nℤ → 𝑪 = ℤ/Nℤ given by f(P) = $\mathcal{C}$ ≡ aP + b (mod N), where a ∈ (ℤ/Nℤ)* and b ∈ ℤ/Nℤ are fixed. The pair K _E = (a, b) is called the enciphering key. To compute f−1, in order to decipher, one needs to compute the deciphering key K _D = (a⁻¹, −a⁻¹b) so that P ≡ a⁻¹C − a⁻¹b (mod N). The keys K _E and K _D are concealed. In the above example and other classical cryptosystems it is possible to determine the deciphering key from the enciphering key (and vice-versa).