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Passphrases vs Random Passwords: The Entropy Math

Six random words from a 7,776-word list give about 77.5 bits of entropy, as strong as a 13-character random password and much easier to remember.

By Muhammad Ahmad. Published . 1 min read.

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A passphrase is several random words strung together. Done properly, it can be as strong as a long random password while being far easier to remember and type.

The math

Entropy counts the choices made at random. For a password, each character is one choice from the character set. For a passphrase, each word is one choice from a word list.

  • A standard diceware list has 7,776 words, so each random word adds log₂ 7,776 ≈ 12.9 bits.
  • 4 words ≈ 51.7 bits, 5 words ≈ 64.6 bits, 6 words ≈ 77.5 bits, 7 words ≈ 90.5 bits.

Compared with random characters

A random letters-and-digits password carries about 5.95 bits per character, so 6 random words (77.5 bits) roughly match a 13-character random password.

A 12-character random password for comparison
  1. Bits per character

    log₂ 62equals5.95

  2. Entropy

    12 × 5.95equals71.5 bits

  3. Average time

    62¹² ÷ 2 ÷ 10,000,000,000/sequals5,112 years

The catch: the words must be random

The math only holds if the words are picked at random, by dice or a generator, not chosen by you. A phrase from a song, a quote or a sentence you made up has much less entropy, because attackers try common phrases first.

Tip: Use a generator or real dice to pick the words, and separate them with spaces or dashes you'll remember. Capitalization tricks add little compared with one extra word.

Sources

  1. NIST SP 800-63B: Digital Identity Guidelines, Authentication