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Password Entropy Calculator

See how strong a random password or passphrase is, in bits of entropy, from its length and character sets. It also estimates how long it would take to guess.

Level: Grades 11–12

  • Common Core standard HSF.BF.B.5(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Printable worksheet

Password Entropy Calculator

Inputs
Password style

Result

Entropy

71.45 bits

12 random characters from a pool of 62 give about 71.5 bits of entropy.

Character pool size
62
Bits per character
5.9542
Possible passwords (2^H)
3.23 × 10^21
Average time to guess at 10,000,000,000 guesses/s
about 5,112 yearsAssumes an attacker has a copy of a fast, unprotected hash (the scrambled form a site stores) and guesses offline. See the table for other assumptions.

See the step-by-step solution ↓

Time to guess under different assumed attack speeds
Time to guess under different assumed attack speeds
Assumed scenarioGuesses per secondAverage timeWorst case (whole space)
Online login, lightly throttled100about 511,171,122,392 yearsabout 1,022,342,244,783 years
Offline, deliberately slow hash10,000about 5,111,711,224 yearsabout 10,223,422,448 years
Offline, fast hash on GPUs10,000,000,000about 5,112 yearsabout 10,223 years
Offline, very large cluster1,000,000,000,000about 51.1 yearsabout 102 years

Step-by-step solution

  1. Add up the sizes of the chosen sets (lowercase, uppercase, digits) to get the pool size

    N=26+26+10=62

  2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

    log₂(62)=5.954196 bits

  3. Multiply by the number of characters, because every random pick adds the same bits. This gives the entropy H

    H=12×5.954196=71.450356 bits

  4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

    271.452×10,000,000,000=161,313,338,119.8938 s=about 5,112 years

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

See how hard a randomly generated password would be to guess. Password entropy measures how many guesses an attacker would need, counted in bits. Each extra bit doubles the number of possibilities. A password with 40 bits of entropy has about a trillion possible values. One with 80 bits has about a trillion times more than that. Entropy is the fairest way to compare a long passphrase with a short jumble of symbols.

This calculator works from the recipe, not the password. Tell it how long the password is and which character sets it draws from. Or tell it how many words a passphrase uses and how big the word list is. It never asks for the password itself, and you shouldn't type a real password into any website to test it. There's a math reason too. Entropy belongs to the way a password was made, not to the characters it happens to contain.

Where it's used

Why this math matters outside the classroom:

  • Random passwords. A 12-character password drawn from 94 printable characters has 12 × log₂ 94 ≈ 78.7 bits of entropy.
  • Passphrases. A passphrase of 6 words picked at random from a 7,776-word list has about 6 × 12.9 ≈ 77.5 bits.
  • Length rules. IT teams set minimum length rules after seeing that each extra character multiplies the guesses needed.
  • Doubling guesses. Students learn that each extra bit of entropy doubles the number of guesses an attacker needs on average.

Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSF.BF.B.5. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose random characters or a random-word passphrase.
  2. For characters, enter the length and tick each set the generator draws from. For a passphrase, enter the number of words and the size of the word list.
  3. Read the entropy in bits. Then check the time-to-guess table, which shows how the answer changes at different attack speeds.

What the inputs mean

Length
The number of characters. Each one should be picked on its own, at random, from the whole pool.
Symbols (32)
The 32 punctuation characters on a standard US keyboard, such as ! ? # and @. Together the four sets make 94 printable characters.
Number of words
How many words are picked, each one on its own, at random.
Word-list size
The number of words the generator chooses from. Lists designed for dice have 7,776 words (five dice, 6⁵ outcomes).

Formula and method

N = size of the pool (characters or words)
H = L × log₂(N)  bits
possible passwords = N^L = 2^H
average guessing time = 2^H ÷ 2 ÷ guesses per second

N is the size of the pool, L is the length (characters or words) and H is the entropy in bits. log₂(N) asks: how many times would you double 1 to reach N? For example, log₂(8) = 3, because 2 × 2 × 2 = 8.

Say every symbol is picked at random from N options, each equally likely. Then each one adds log₂(N) bits, and L separate picks add L times that. The total is how many doublings it takes to reach the number of possible passwords.

An attacker who tries every possibility will, on average, find the right one halfway through. So the average time uses half of all the possibilities, and the worst case uses all of them. The guess rates in the table are stated assumptions, from very slow to very fast. Real speeds depend on how the password is stored. Sites usually store a hash, a scrambled version of the password that's hard to reverse.

Worked example

A password manager generates a 12-character password from lowercase, uppercase and digits.

  1. Pool size: 26 + 26 + 10 = 62 characters.
  2. Bits per character: log₂(62) = 5.9542.
  3. Entropy: 12 × 5.9542 = 71.45 bits, or about 3.23 × 10^21 possible passwords.
  4. At an assumed 10 billion guesses per second, the average search covers half of that: about 5,112 years.

Result: The password has about 71.45 bits of entropy. For comparison, a six-word passphrase from a 7,776-word list reaches 77.55 bits.

Understanding the result

Every added character or word multiplies the attacker's work. Adding a character set raises it more slowly. Going from 12 to 16 characters with the same 62-character pool adds about 24 bits. Adding symbols to a 12-character password adds only about 7.

The table shows why the way a password is stored matters as much as the password. Against a fast, unprotected hash, a password might fall in hours. Against a hash that's slow on purpose, the same one can take far longer than a lifetime. And an online login that limits attempts slows attackers even more.

Assumptions and limitations

  • A good generator picks every character or word on its own, at random, with each option equally likely.
  • The attacker knows the recipe (length, sets or list) but not the password. That's the standard, cautious assumption.
  • Guess rates in the table are example assumptions, not measurements of any particular system.
  • The formula overstates the strength of anything a person made up. Real attackers try words, names, dates, keyboard patterns and predictable swaps early. So a password a person chose can be far weaker than its length suggests.
  • Entropy says nothing about reuse, phishing or leaks. A strong password reused on a site that gets breached offers no protection elsewhere.
  • Times are averages for trying every possibility. An attacker can get lucky early.

Frequently asked questions

Why doesn't this calculator let me type my password?

Two reasons. First, you should never enter a real password into a web page that doesn't need it. Second, the number you want depends on how the password was made. "Summer2024!" looks like 11 characters from 94 symbols, about 72 bits. But it follows a pattern attackers try early, so its real strength is far lower. Describing the generator gives an honest answer; looking at the password itself doesn't.

How many bits of entropy is enough?

It depends on the threat. As a rough guide, take a password where every symbol is truly random. Above about 70–80 bits, an offline search is out of reach of large attackers for the foreseeable future. 128 bits matches the strength of modern encryption keys. Accounts protected by attempt limits and a second login step can get by with less.

Is a passphrase better than a random string of symbols?

Character for character, random symbols carry more entropy. But passphrases are much easier to remember and type. Six random words from a 7,776-word list give about 77.5 bits. That's close to a 12-character password (78.7 bits) drawn from all 94 printable characters.

Does replacing letters with numbers or symbols help?

Very little when a person does it. Swaps like a → @ or o → 0 are well known, and attackers try them automatically. Length chosen at random adds far more strength than predictable decoration.

How this calculator is checked

Method: Information entropy of a uniformly random choice (H = L × log₂ N); exhaustive-search expected time of half the key space. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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