Editing Cryptosystem

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{{More categories|date=January 2025}}{{short description|Suite of cryptographic algorithms needed to implement a particular security service}}
In [[cryptography]], a '''cryptosystem''' is a suite of [[cryptographic algorithm]]s needed to implement a particular security service, such as confidentiality ([[encryption]]).<ref>{{cite book|last1=Menezes|first1=A.|last2=Oorschot|first2=P. van|last3=Vanstone|first3=S.|title=Handbook of Applied Cryptography|year=1997|publisher=CRC Press|isbn=0-8493-8523-7|edition=5th|url-access=registration|url=https://archive.org/details/handbookofapplie0000mene}}</ref>

Typically, a cryptosystem consists of three algorithms: one for [[key generation]], one for encryption, and one for decryption. The term ''[[cipher]]'' (sometimes ''cypher'') is often used to refer to a pair of algorithms, one for encryption and one for decryption. Therefore, the term ''cryptosystem'' is most often used when the key generation algorithm is important. For this reason, the term ''cryptosystem'' is commonly used to refer to [[public key cryptography|public key]] techniques; however both "cipher" and "cryptosystem" are used for [[Symmetric-key algorithm|symmetric key]] techniques.

== 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.

== Examples ==

A classical example of a cryptosystem is the [[Caesar cipher]]. A more contemporary example is the [[RSA (cryptosystem)|RSA]] cryptosystem.

Another example of a cryptosystem is the [[Advanced Encryption Standard]] (AES). AES is a widely used symmetric encryption algorithm that has become the standard for securing data in various applications.

[[Paillier cryptosystem]] is another example used to preserve and maintain privacy and sensitive information. It is featured in electronic voting, electronic lotteries and electronic auctions.<ref>{{Cite book |last1=Xia |first1=Zhe |last2=Yang |first2=Xiaoyun |last3=Xiao |first3=Min |last4=He |first4=Debiao |chapter=Provably Secure Threshold Paillier Encryption Based on Hyperplane Geometry |series=Lecture Notes in Computer Science |date=2016 |volume=9723 |editor-last=Liu |editor-first=Joseph K. |editor2-last=Steinfeld |editor2-first=Ron |title=Information Security and Privacy |chapter-url=https://link.springer.com/chapter/10.1007/978-3-319-40367-0_5 |language=en |location=Cham |publisher=Springer International Publishing |pages=73–86 |doi=10.1007/978-3-319-40367-0_5 |isbn=978-3-319-40367-0}}</ref>

==See also==
*[[List of cryptosystems]]
*[[Semantic security]]

== References ==
{{Reflist}}

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[[Category:Cryptography]]