# Cryptosystem

__: Suite of cryptographic algorithms needed to implement a particular security service__

**Short description**In cryptography, a **cryptosystem** is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption).^{[1]}

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 techniques; however both "cipher" and "cryptosystem" are used for symmetric key techniques.

## Formal definition

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

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

For each [math]\displaystyle{ e \in \mathcal{K} }[/math], there is [math]\displaystyle{ d \in \mathcal{K} }[/math] such that [math]\displaystyle{ D_d(E_e(p)) = p }[/math] for all [math]\displaystyle{ p \in \mathcal{P} }[/math].^{[2]}

Note; typically this definition is modified in order to distinguish an encryption scheme as being either a symmetric-key or public-key type of cryptosystem.

## Examples

A classical example of a cryptosystem is the Caesar cipher. A more contemporary example is the RSA cryptosystem.

## References

- ↑ Menezes, A.; Oorschot, P. van; Vanstone, S. (1997).
*Handbook of Applied Cryptography*(5th ed.). CRC Press. ISBN 0-8493-8523-7. https://archive.org/details/handbookofapplie0000mene. - ↑ Buchmann, Johannes A. (13 July 2004).
*Introduction to Cryptography*(2nd ed.). Springer. ISBN 0-387-20756-2.

Original source: https://en.wikipedia.org/wiki/Cryptosystem.
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