Password Entropy, Explained: Why Length Beats Complexity

Password strength has a real unit of measurement: entropy, expressed in bits. Entropy measures how many equally likely possibilities an attacker must search, and each additional bit doubles that number. For a password of random characters, the formula is simple: entropy = length × log₂(alphabet size). Lowercase letters give log₂(26) ≈ 4.7 bits per character; the full keyboard of ~95 printable characters gives log₂(95) ≈ 6.6. A truly random 12-character full-keyboard password is therefore about 79 bits — a number so large that guessing becomes physically impractical.

The catch: entropy assumes random

The formula only holds for passwords chosen uniformly at random. Human-chosen passwords are nothing like random — they're words, names, dates, and predictable substitutions, and cracking software tries exactly those patterns first. P@ssw0rd! looks like it uses a 95-character alphabet, but it's a dictionary word with the four most common substitutions and the most common suffix position, and it falls almost instantly. This is why NIST's Digital Identity Guidelines (SP 800-63B) moved away from composition rules ("must contain an uppercase, a number, a symbol") and toward length, screening against known-breached passwords, and encouraging passphrases: mandatory complexity produces predictable patterns, while length produces real entropy.

How long passwords actually survive

The worst case is an offline attack: an attacker has stolen a password database and can test guesses on their own hardware. Modern GPU rigs testing a fast-hashed database can plausibly try on the order of a trillion (10¹²) guesses per second. Average time to find a random password at that speed (half the keyspace):

Two things jump out. First, the cliff between 8 and 16 characters: doubling the length doesn't double the strength, it raises it by 47 bits — a factor of about 10¹⁴. Second, one extra passphrase word multiplies the work by ~7,776. (Online attacks against a website's login form are millions of times slower because sites rate-limit attempts — the table is the stolen-database scenario, which is the one your password's strength actually has to survive.)

Why word lists work: Diceware and the EFF list

A passphrase like gravel-pony-uneven-slang-crayon is easier to remember than kX9#mQ2v and dramatically stronger — provided the words are chosen at random. The EFF's Large Wordlist contains 7,776 words (chosen so five dice rolls can pick one), so each word contributes log₂(7776) ≈ 12.9 bits even though the attacker has the entire list. That's the counterintuitive core of the method: the security lives in the random selection, not in the secrecy of the words. Picking words yourself defeats it — human word choices are as predictable as human passwords.

What to actually do

Use a password manager and let it generate long random passwords — 16+ characters — for every account, so no two sites share a password and a breach at one can't cascade. For the handful of secrets you must remember (the manager's master password, your computer login), use a randomly generated passphrase of five or six words. And turn on multi-factor authentication where offered: it protects you even on the day a password does leak.

The Password Generator builds both kinds — random character passwords and EFF-wordlist passphrases — using your browser's cryptographic random number generator, shows the entropy math for each, and never sends anything over the network.

Try it yourself

This guide pairs with the free Password Generator — no sign-up, runs in your browser, and shows its formula.