Last updated: 2026-06-23

Password Entropy

Password entropy measures how unpredictable a password is, expressed in bits. It determines the minimum number of guesses an attacker must make in a brute-force search: a password with $H$ bits of entropy forces the attacker to test up to $2^H$ candidates in the worst case, and $2^H / 2$ on average. Entropy is the standard metric used by the National Institute of Standards and Technology (NIST) when evaluating password strength.

Formula

When a password of length $L$ is drawn uniformly at random from a character pool of size $N$, there are $N^L$ equally likely passwords. The entropy in bits is the base-2 logarithm of that count:

The expected crack time for an attacker testing $G$ passwords per second is:

The denominator of 2 converts worst-case to median: in a random search, the correct password is found after testing half the space on average.

Worked Example

A 12-character password drawn from the 62-character alphanumeric pool (a–z, A–Z, 0–9):

Against a fast offline attack at $10^{10}$ guesses per second — typical for a modern GPU cracking MD5:

That same password faces a very different timeline if the attacker is limited to an online login service rate-limited to 1,000 guesses per second, where the expected crack time exceeds $5 \times 10^{13}$ years. Conversely, a future massively parallel cluster at $10^{12}$ guesses per second reduces the estimate to roughly 51 years — still practically infeasible, but illustrating why higher-stakes systems use slow password hashing algorithms such as bcrypt or Argon2.

Factors That Determine Entropy

Character pool size contributes logarithmically. Moving from lowercase only (26 characters) to all printable ASCII (94 characters) on a fixed 12-character password adds approximately 21 bits of entropy:

$12 \times (\log_2 94 - \log_2 26) \approx 21 \text{ bits}$

Length ($L$) contributes linearly. Each additional character adds $\log_2(N)$ bits — about 6 bits per character for the alphanumeric pool.

Because length scales linearly and pool size scales only logarithmically, lengthening a password is generally more efficient than adding character classes. A 20-character lowercase passphrase reaches roughly 94 bits — stronger than a 12-character full-ASCII password at 79 bits.

Limitations

Entropy calculations assume the password is generated uniformly at random. Human-chosen passwords are not random: they cluster around patterns (dictionary words, keyboard walks, common substitutions) that attackers exploit before resorting to pure brute force. A 12-character password that spells "Password1" has 71 bits of theoretical entropy but can be found very rapidly because it matches well-known patterns. Use a cryptographically random password generator or passphrase tool to achieve the entropy this calculator reports.

For a related analysis of hash collision probability, see Hash Collision Probability Calculator.