What is password entropy, really?

Entropy is the single number that tells you how hard a password is to guess. Strip away the jargon and it's just counting — here's the whole idea from first principles.

When we say a password has "128 bits of entropy," we mean an attacker who knows exactly how it was made would still have to try, on average, half of 2^128 possibilities to land on it. Entropy measures unpredictability — not how exotic the characters look.

The formula

For a password built by drawing each character independently and at random from a pool of R possible symbols, with length L, the entropy in bits is:

Two levers, then: the pool size R and the length L. Each extra character adds a fixed log₂(R) bits, which is why length is the most reliable way to buy strength.

• Lowercase only → R = 26 → log₂(26) ≈ 4.7 bits per character
• Upper + lower + digits → R = 62 → ≈ 5.95 bits per character
• Add symbols → R ≈ 95 → ≈ 6.57 bits per character
• Raw hex → R = 16 → exactly 4 bits per character

The crucial caveat: randomness

That formula only holds when every character is chosen independently and uniformly at random. The moment a human picks the characters, the assumption collapses. P@ssw0rd1 draws from a 95-symbol pool across 9 characters — about 59 "theoretical" bits — yet a cracker breaks it instantly, because it's a dictionary word with predictable substitutions. The real entropy is close to zero.

What the numbers buy you

Assume a well-resourced offline attacker at 10^11 guesses per second against fast-hashed or unsalted data. Rough resistance by entropy:

A practical target: aim for 75+ bits for an account password and 128 bits for anything machine-to-machine, like an API key or session secret.

Mapping it back to the tools

• Length beats complexity. Going from 12 to 16 random characters adds more entropy than sprinkling in symbols ever will.
• Bigger pool, fewer characters needed. A 64-symbol URL-safe token packs 6 bits per character, so 22 characters clears 128 bits.
• Hex is honest but verbose. At 4 bits per character you need 32 hex characters for a 128-bit key — which is exactly why 256-bit keys print as 64 hex digits.