How Strong Passwords Work: Length, Entropy and Passphrases

"Use a capital, a number and a symbol" is decades-old advice that misses the point. What makes a password strong is how unpredictable it is and how long. Here is the maths, in plain terms.

Entropy: the strength measure

Password strength is usually expressed in bits of entropy. Each bit doubles the number of guesses an attacker needs. A password with 40 bits of entropy has about a trillion possibilities; one with 80 bits has about a million million million million (10²⁴).

For a password made of characters picked truly at random from a set, entropy is:

entropy (bits) = length × log₂(size of character set)

• Lowercase letters only (26): about 4.7 bits per character.
• Lowercase + uppercase + digits (62): about 5.95 bits per character.
• All printable keyboard characters (about 94): about 6.55 bits per character.

A 12-character random password from the full set has roughly 79 bits. A 16-character one has roughly 105 bits. Both are far beyond what any attacker can brute-force against a properly protected account.

Why length beats complexity

Adding a character multiplies the search space by the size of the character set. Adding a symbol type adds only a little to that size. Compare: an 8-character password using all 94 characters has about 52 bits; a 12-character one using lowercase only has about 56 bits. The longer plain one is already stronger.

The catch in the formula is the word random. Passwords humans invent, such as "P@ssw0rd!" or "Summer2026!", follow predictable patterns that cracking software tries first. Their real strength is a tiny fraction of what the formula suggests.

How passwords actually get cracked

• Credential stuffing: breached passwords from one site tried on others. This is by far the biggest danger, and it is why reuse is the worst habit.
• Dictionary and pattern attacks: common words, names, dates, keyboard patterns and predictable substitutions (a→@, o→0).
• Brute force against a stolen hash: when a site's database leaks, attackers guess offline at billions per second against weak hashing. Long random passwords make this impractical.
• Phishing and malware: no password strength helps if you type it into a fake site. Passkeys and two-factor authentication do.

Passphrases

A passphrase is several random words joined together. The important word again is random. Choosing four words from a list of 7,776 (the Diceware list) gives 4 × log₂(7776) ≈ 51.7 bits; five words ≈ 64.6 bits; six words ≈ 77.5 bits. Six random words are comparable to a 12-character random password, and much easier to type and remember.

A quote, lyric or phrase you pick yourself is not a random passphrase, and it has far less entropy than its length suggests.

What to do in practice

1. Use a password manager and let it generate a unique random password, 16 characters or more, for every account. You only need to remember one strong master passphrase.
2. Use a random passphrase of six or more words for the few things you must type from memory: your manager's master password, your computer login.
3. Turn on two-factor authentication, preferably an authenticator app or a passkey, for email, banking and anything important.
4. Never reuse passwords. If you do only one thing, do this.
5. Ignore forced rotation unless you suspect a breach. Current guidance from NIST recommends against periodic changes and against composition rules, favouring length and breach checking.

Generating passwords safely

A generator is only as good as its randomness. Ours uses your browser's crypto.getRandomValues(), the same cryptographic source used for encryption keys, and picks characters without statistical bias. The password is created on your device and never transmitted or stored. Length, character types and an option to exclude look-alike characters such as O/0 and l/1 are all adjustable.

Frequently asked questions

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