How Strong Is Your Password? The Math Behind Password Strength
Password strength isn't a feeling — it's entropy. Learn how character sets, length and randomness interact, why length beats complexity, and how attackers actually guess.
“This password is pretty strong, I think.” Almost everyone says this. Almost no one can back it up. Password strength isn’t an impression — it’s a number you can calculate, and once you can calculate it, a lot of common advice starts making sense (and some of it starts looking silly).
Strength = entropy, measured in bits
The strength of a randomly generated password is its entropy:
Entropy (bits) = log₂(N^L) = L × log₂(N)
Where N is the size of the character set and L is the length.
- Lowercase letters only: N = 26
- Lowercase + uppercase: N = 52
-
- digits: N = 62
-
- symbols: N = 94 (roughly)
Some real numbers:
| Password | Charset | Length | Entropy |
|---|---|---|---|
Tr0ub4dor&3 | 94 | 11 | ~72 bits* |
| Random 12 chars (all sets) | 94 | 12 | ~79 bits |
| Random 16 chars (all sets) | 94 | 16 | ~105 bits |
| 6 random words | 7,776/word | 6 | ~77 bits |
*The famous xkcd example — but only if the substitutions are random. If a human chose them, the real entropy is far lower, because humans are predictable. Tr0ub4dor&3 is a dictionary word with predictable substitutions; cracking tools try exactly those patterns first.
The key insight: length beats complexity
Look at the math. Each extra character from a 62-character set multiplies the search space by 62 (adds ~6 bits). Upgrading your character set only changes the per-character bonus once.
- Going from 8 to 12 characters (lowercase only): 37.6 → 56.4 bits (+18.8)
- Going from lowercase-only to full mixed sets at 8 characters: 37.6 → 47.6 bits (+10)
Doubling the “complexity” adds less than a third of what adding four plain characters does. This is why the modern consensus (NIST, and every security team worth reading) is: long and memorable beats short and gnarly.
Four random words — copper-notebook-drift-lantern — have ~51 bits at minimum with a small wordlist, and 70+ with a larger one, while being something you can actually type on a phone.
How attackers actually guess
Offline cracking speed against a stolen password database is measured in billions of guesses per second on commodity GPU hardware. What that means for entropy:
- 40 bits: hours
- 60 bits: years
- 80+ bits: not brute-forceable; attackers move on to other tricks
But note the phrase “offline cracking.” Most real-world account takeovers don’t involve brute force at all. They use:
- Credential stuffing — replaying passwords stolen from other sites (the reason password reuse is the single worst habit)
- Targeted dictionaries — names, dates, sports teams,
Password1!and every leetspeak variant - Phishing — no math required
Entropy protects you against the first scenario only when your password is random. Against reuse and phishing, entropy does nothing — a password manager and unique passwords do.
Three rules that actually matter
- Unique per site, always. One breach shouldn’t unlock your whole life.
- Length first. 14+ random characters, or 4+ random words, for anything important.
- Randomness is the non-negotiable part. “Correct horse battery staple” typed by you isn’t random — you chose those words because they felt nice, and attackers model exactly that feeling. Use a generator with a cryptographic random source.
Check yours right now
The Password Generator creates passwords with a cryptographic random source, shows the entropy in bits as you adjust length and character sets, and copies with one click. Generate a 16-character password, note the number (~105 bits), then try to construct one by hand that you’d trust the same way.
And a small related tip: some systems have length limits or reject certain characters — the Word Counter counts characters instantly, which is handy when a form says “maximum 64 characters” and your passphrase is sitting in your notes.