Brute Force Time (Charset × Length)
Estimate how long an exhaustive password search takes given alphabet size, length, and guess rate.
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The engineering
The keyspace of a password is the alphabet size C raised to the length L. Multiply by the fraction of that space an attacker expects to cover (0.5 on average to hit a randomly-placed target) and divide by the guess rate r to get wall-clock time. Entropy in bits, H = L·log₂C, is the same quantity on a log-2 scale and is what NIST uses to reason about password strength.
The dominant lever is length, not charset: adding one character multiplies the keyspace by C, while jumping from lowercase-only to full mixed-case-plus-symbols only roughly doubles it. The gotcha is the guess rate — an online login throttled to a few per second is worlds apart from an offline GPU cracking rig doing 10⁹–10¹² unsalted MD5 hashes per second. This card assumes true random passwords; human-chosen ones fall far faster because the effective keyspace is tiny.
Where this math comes from
The idea that password strength is exhaustive-search cost goes back to the early Multics and Unix days: Robert Morris and Ken Thompson's 1979 paper 'Password Security: A Case History' laid out why crypt() was slow on purpose and why salting mattered — brute force was already the threat model. They measured that most user passwords were guessable in hours even then.
NIST formalized the bookkeeping in SP 800-63 (2004), quantifying password guessing entropy in bits and prescribing rate limits. The 2017 revision SP 800-63B threw out the old character-composition rules and told everyone to favor length and blocklists instead — an acknowledgment that theoretical keyspace and real-world guessability are not the same number.
- 1979Robert Morris & Ken ThompsonPublish the Unix password study — deliberately slow hashing and salting to raise brute-force cost.
- 1988Alec MuffettReleases Crack, popularizing offline dictionary and brute-force password auditing.
- 2004NIST (SP 800-63)Standardizes password entropy accounting in bits.
- 2017NIST (SP 800-63B)Prioritizes length and blocklists over composition rules.
See the full timeline of the math behind every calculator →
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