Editing Cryptosystem (section)

== Formal definition ==

Mathematically, a cryptosystem or encryption scheme can be defined as a [[tuple]] <math>(\mathcal{P},\mathcal{C},\mathcal{K},\mathcal{E},\mathcal{D})</math> with the following properties.

#<math> \mathcal{P}</math> is a set called the "plaintext space". Its elements are called plaintexts.
#<math> \mathcal{C}</math> is a set called the "ciphertext space". Its elements are called ciphertexts.  
#<math> \mathcal{K}</math> is a set called the "key space". Its elements are called keys.
#<math> \mathcal{E} = \{ E_k : k \in \mathcal{K} \} </math> is a set of functions <math>E_k : \mathcal{P} \rightarrow \mathcal{C}</math>. Its elements are called "encryption functions".
#<math> \mathcal{D} = \{ D_k : k \in \mathcal{K} \} </math> is a set of functions <math>D_k : \mathcal{C} \rightarrow \mathcal{P}</math>. Its elements are called "decryption functions".

For each <math>e \in \mathcal{K}</math>, there is <math>d \in \mathcal{K}</math> such that <math>D_d(E_e(p)) = p</math> for all <math>p \in \mathcal{P}</math>.<ref>{{cite book|last1=Buchmann|first1=Johannes A.|title=Introduction to Cryptography|date=13 July 2004|publisher=Springer|isbn=0-387-20756-2|edition=2nd}}</ref>

Note; typically this definition is modified in order to distinguish an encryption scheme as being either a [[Symmetric-key algorithm|symmetric-key]] or [[Public-key cryptography|public-key]] type of cryptosystem.