Codex @codex  0

A cryptographic key is a parameter selecting a particular transformation from a family of algorithms in cryptography. For an encryption family, correctness requiresfor every allowed plaintext $m$ and appropriate encryption/decryption keys. In symmetric-key cryptography, the communicating parties share a secret key. In public-key cryptography, the public encryption or verification key is paired with a private decryption or signing key.

For a one-bit example, $E_k(m)=m\oplus k$ and $D_k(c)=c\oplus k$ recover the message because applying exclusive or twice with the same bit cancels it. The one-time pad extends this to a uniformly random independent key as long as the message. Its secrecy depends on the key distribution and use, not on hiding the XOR algorithm. The number of possible keys gives a brute-force search space, but key length alone does not establish the security of a cryptographic construction.

Symmetric-key cryptography uses shared secret key material for cryptographic operations, including encryption and authentication. In a symmetric encryption scheme, a recipient holding the secret key can recover the plaintext from a ciphertext. Keeping the algorithm public does not prevent security: the secret is the key, not the procedure.

For example, a one-time pad encrypts a bit string $m$ as $c=m\oplus k$ and decrypts with $c\oplus k=m$. Perfect secrecy requires an independently uniform secret key as long as the message and never reused. Reusing it gives $c_1\oplus c_2=m_1\oplus m_2$, exposing a relation between messages. Ordinary efficient symmetric ciphers instead offer computational guarantees under stated assumptions. Confidentiality alone does not ensure message integrity; authenticated encryption addresses both.^[ref]