ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D468
Bijective map

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
Children
 ▶ D353: Set automorphism ▶ D2221: Set of bijections
Results
 ▶ R2764: Canonical singleton map is bijection ▶ R1841: Indicator function operator is a bijection ▶ R4540: Inverse map is a bijection ▶ R5097: Power set is isomorphic to a set of boolean functions
Examples
 ▶ 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 $$e_k : P \to C$$ such that $e_k(p)$ is a ciphertext for any $p \in P$. The corresponding decryption map would then be the map $$d_k : C \to P$$ such that $$d_k(e_k(p)) = p$$ 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.