In cryptography, a **cryptosystem** is a suite of cryptographic algorithms needed to implement a particular security service, most commonly for achieving 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 $({\mathcal {P}},{\mathcal {C}},{\mathcal {K}},{\mathcal {E}},{\mathcal {D}})$ with the following properties.

- ${\mathcal {P}}$ is a set called the "plaintext space". Its elements are called plaintexts.
- ${\mathcal {C}}$ is a set called the "ciphertext space". Its elements are called ciphertexts.
- ${\mathcal {K}}$ is a set called the "key space". Its elements are called keys.
- ${\mathcal {E}}=\{E_{k}:k\in {\mathcal {K}}\}$ is a set of functions $E_{k}:{\mathcal {P}}\rightarrow {\mathcal {C}}$. Its elements are called "encryption functions".
- ${\mathcal {D}}=\{D_{k}:k\in {\mathcal {K}}\}$ is a set of functions $D_{k}:{\mathcal {C}}\rightarrow {\mathcal {P}}$. Its elements are called "decryption functions".

For each $e\in {\mathcal {K}}$, there is $d\in {\mathcal {K}}$ such that $D_{d}(E_{e}(p))=p$ for all $p\in {\mathcal {P}}$.^{[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. *Handbook of Applied Cryptography* (5th ed.). CRC Press. ISBN 0-8493-8523-7.
**^** Buchmann, Johannes A. *Introduction to Cryptography* (2nd ed.). Springer. ISBN 0-387-20756-2.