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.

Putting entropy theory into practice means using tools that generate truly random passwords and store them safely. NordPass fills both roles, it generates cryptographically random passwords with measurable entropy and keeps them in an encrypted vault you can access across all your devices.