Cryptosystem

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.

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So the first main topic we'll introduce is symmetric cryptosystems. Symmetric means both encryption and decryption are done with the same key. I want to introduce some terminology that we'll use throughout the course. So we can think of encryption as a function, and it takes as input a plain text--this is the unencrypted message-- it outputs a ciphertext, and then our goal is to be able to send that ciphertext over an insecure channel. This could be a wireless network, this could be the internet, this could be a courier carrying your message, any channel where we can't trust the channel to be secure. We hope what comes out of the channel is the same ciphertext we put into it. Then that goes into a decryption function, and what comes out is the message that we sent it. Our cast of characters--we often talk about Alice sending the message, Bob receiving it. We also have malicious characters who might be listening in on this channel. One of those is called Eve--for eavesdropper--and she has really big ears and can hear the messages sent between Alice and Bob, but she only hears what's sent over the insecure channel, she doesn't have access to the plain text--that's the input to the encryption function-- or the plain text that comes out of the decryption function at the other end. But she can hear the ciphertext that's sent over the insecure channel. To think about this a little more precisely, the plain text is some message, and it's selected from a set of messages. So, M is the set of all possible messages, and M could be finite--for a given length, M is finite-- and we'll often be thinking about fixed-length messages, so there's a finite set of them, and M is some message selected from that set. The ciphertext is C--selected from the set of all possible ciphertexts, and the encryption and decryption functions are functions. The encryption function--as shown here-- is a function that goes from an element of M to an element of C. The decryption function goes from an element of C--a ciphertext--to an element of M. In order for Bob to receive the same message as the one Alice encrypted, we need this property--that the D function is an inverse of the M function. So we need it to be the case for all messages. The result of decryption the encryption of that message is the same as the message we started with.

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.

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.

Pallier cryptosystem is another example used to preserve and maintain privacy and senstive information. It is featured in electronic voting, electronic lotteries and electronic auctions.^[3]

References

^ Menezes, A.; Oorschot, P. van; Vanstone, S. (1997). Handbook of Applied Cryptography (5th ed.). CRC Press. ISBN 0-8493-8523-7.
^ Buchmann, Johannes A. (13 July 2004). Introduction to Cryptography (2nd ed.). Springer. ISBN 0-387-20756-2.
^ Xia, Z., Yang, X., Xiao, M., & He, D. (n.d.). Provably Secure Threshold Paillier Encryption Based on Hyperplane Geometry. In Information Security and Privacy (pp. 73–86). Springer International Publishing. https://doi.org/10.1007/978-3-319-40367-0_5

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General	History of cryptography Outline of cryptography Cryptographic protocol Authentication protocol Cryptographic primitive Cryptanalysis Cryptocurrency Cryptosystem Cryptographic nonce Cryptovirology Hash function Cryptographic hash function Key derivation function Digital signature Kleptography Key (cryptography) Key exchange Key generator Key schedule Key stretching Keygen Cryptojacking malware Ransomware Random number generation Cryptographically secure pseudorandom number generator (CSPRNG) Pseudorandom noise (PRN) Secure channel Insecure channel Subliminal channel Encryption Decryption End-to-end encryption Harvest now, decrypt later Information-theoretic security Plaintext Codetext Ciphertext Shared secret Trapdoor function Trusted timestamping Key-based routing Onion routing Garlic routing Kademlia Mix network
Mathematics	Cryptographic hash function Block cipher Stream cipher Symmetric-key algorithm Authenticated encryption Public-key cryptography Quantum key distribution Quantum cryptography Post-quantum cryptography Message authentication code Random numbers Steganography
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Formal definition

Examples

See also

References