Cryptography/Symmetric Ciphers

A symmetric key cipher (also called a secret-key cipher, or a one-key cipher, or a private-key cipher, or a shared-key cipher) Shared_secretis one that uses the same (necessarily secret) key to encrypt messages as it does to decrypt messages.

Until the invention of asymmetric key cryptography (commonly termed "public key / private key" crypto) in the 1970s, all ciphers were symmetric. Each party to the communication needed a key to encrypt a message; and a recipient needed a copy of the same key to decrypt the message. This presented a significant problem, as it required all parties to have a secure communication system (e.g. face-to-face meeting or secure courier) in order to distribute the required keys. The number of secure transfers required rises impossibly, and wholly impractically, quickly with the number of participants.

Formal Definition


Any cryptosystem based on a symmetric key cipher conforms to the following definition:

  • M : message to be enciphered
  • K : a secret key
  • E : enciphering function
  • D : deciphering function
  • C : enciphered message. C := E(M, K)
  • For all M, C, and K, M = D(C,K) = D(E(M,K),K)

Reciprocal Ciphers


Some shared-key ciphers are also "reciprocal ciphers." A reciprocal cipher applies the same transformation to decrypt a message as the one used to encrypt it. In the language of the formal definition above, E = D for a reciprocal cipher.

An example of a reciprocal cipher is Rot 13, in which the same alphabetic shift is used in both cases.

The xor-cipher (often used with one-time-pads) is another reciprocal cipher.[1]

Reciprocal ciphers have the advantage that the decoding machine can be set up exactly the same as the encoding machine -- reciprocal ciphers do not require the operator to remember to switch between "decoding" and "encoding".[2]

Symmetric Cypher Advantages


Symmetric key ciphers are typically much less computational overhead Overhead_(computing) than Asymmetric ciphers, sometimes this difference in computing overhead per character can be several orders of magnitude[3]. As such they are still used for bulk encryption of files and data streams for online applications.

To set up a secure communication session Session_key between two parties the following actions take place Transport_Layer_Security:

  1. Alice tells Bob (in cleartext) that she wants a secure connection.
  2. Bob generates a single use (session), public/private (asymmetric) key pair (Kpb Kpr).
  3. Alice generates a single use (session) symmetric key. This will be the shared secret (Ks).
  4. Bob sends Alice the public key (Kpb).
  5. Alice encrypts her shared session key Ks with the Public key Kpb Ck := E(Ks, Kpb) and sends it to Bob
  6. Bob decrypts the message with his private key to obtains the shared session key Ks := D(Ck, Kpr)
  7. Now Alice and Bob have a shares secret (symmetric key) to secure communication on this connection for this session
  8. Either party can encrypt a message simply by C := E(M, Ks) and decrypt is by M = D(C,K) = D(E(M,Ks),Ks)

This is the Basis for Diffie–Hellman Diffie_Hellman_key_exchange key exchange Key exchange and its more advanced successors Transport_Layer_Security.



Further Reading


Symmetric-key algorithm


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  1. Wikipedia: Symmetric-key algorithm#Reciprocal cipher
  2. Greg Goebel. "The Mechanization of Ciphers". 2018.