One-Time Pad

Updated 03 Sep 2026

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  Definition

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    the canonical example of a Shannon Cipher — keys, messages, and ciphertexts are all bit strings of the same fixed length $L$

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    encryption: $E(k, m) := k \oplus m$; decryption: $D(k, c) := k \oplus c$ (XOR is its own inverse)

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  history

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    due to Gilbert Vernam, 1917 — though there's evidence it was discovered earlier

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  variants

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    variable-length one-time pad — keys are a fixed length $L$, but messages/ciphertexts can be any length up to $L$ (using a truncated key); this variant is not perfectly secure, since ciphertext length leaks message length

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    additive one-time pad — defined over $\{0, \ldots, n-1\}$ using addition mod $n$ instead of XOR; still perfectly secure

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    multiplicative one-time pad — defined over $\{1, \ldots, p-1\}$ for prime $p$, using multiplication mod $p$; still perfectly secure

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  key result

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    the (fixed-length) one-time pad is a perfectly secure Shannon cipher — for any message and ciphertext, there's exactly one key connecting them

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  given message and ciphertext, the key is trivial to recover

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    if you're handed both $m$ and $c = E(k, m)$, the key is simply $k = m \oplus c$ — this is fine (the key was only ever meant to be used once), but it's why key reuse is catastrophic; see Two-Time Pad Attack

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  drawback

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    impractical — the key must be as long as the message; see Shannon's Theorem for why this is unavoidable for any perfectly secure cipher

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    if Alice and Bob had a way to securely share a key as long as the message, they could just use that same channel to share the message itself

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  no integrity

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    the one-time pad (and stream ciphers generally) provide zero integrity — see Malleability

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  related

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    Shannon Cipher

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    Perfect Security

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    Shannon's Theorem

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    Two-Time Pad Attack

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    Malleability

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    Stream Cipher

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