ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Definition D468
Bijective map
Formulation 0
A D18: Map $f : X \to Y$ is a bijective map if and only if
(1) $f$ is an D467: Injective map
(2) $f$ is a D466: Surjective map
Set automorphism
Set of bijections
Canonical singleton map is bijection
Indicator function operator is a bijection
Inverse map is a bijection
Power set is isomorphic to a set of boolean functions
Example 1 (Symmetric (or secret key) cryptosystem scheme)
Let $P$ be a set of plaintexts and let $C$ be the set of ciphers. Given a key $k$, an encryption map (or algorithm) is a bijective map \begin{equation} e_k : P \to C \end{equation} such that $e_k(p)$ is a ciphertext for any $p \in P$. The corresponding decryption map would then be the map \begin{equation} d_k : C \to P \end{equation} such that \begin{equation} d_k(e_k(p)) = p \end{equation} for all $p \in P$.

In this kind of a symmetric cryptosystem scheme, both algorithms $e$ and $d$ would be assumed to be known to all parties involved, while the security of the scheme relies on the confidentiality of the $k$, used for both encryption and decryption.