6 min read Technical

Password Entropy Explained: The Science of Password Strength

Password entropy measures the unpredictability of your password in bits. Understanding entropy helps you create passwords that are mathematically proven to resist attacks. Here's the science behind truly secure passwords.

What Is Password Entropy?

Entropy, in information theory, measures randomness or unpredictability. For passwords, entropy quantifies how many guesses an attacker would need to crack your password. Higher entropy means more security.

The Formula

Entropy = log₂(R^L)

Where:

  • R = Size of the character pool (possible characters)
  • L = Length of the password
  • log₂ = Logarithm base 2 (converts to bits)

Simplified: Entropy = L × log₂(R)

Understanding Bits of Entropy

  • 1 bit: 2 possibilities (like a coin flip)
  • 10 bits: 1,024 possibilities
  • 20 bits: ~1 million possibilities
  • 40 bits: ~1 trillion possibilities
  • 60 bits: ~1 quintillion possibilities
  • 80 bits: ~1 septillion possibilities
  • 128 bits: ~340 undecillion possibilities

Calculating Real-World Password Entropy

Character Set Sizes

Character Set Size (R) Bits per Character Example
Numbers only 10 3.32 7294501836
Lowercase only 26 4.70 qwertyzxcv
Lower + Upper 52 5.70 QwErTyZxCv
Alphanumeric 62 5.95 Qw3rTy8xCv
All ASCII printable 95 6.57 Qw3r!y8x@v

Example Calculations

8-character password, all character types (95 chars)

Entropy = 8 × log₂(95) = 8 × 6.57 = 52.6 bits

Time to crack at 1 billion guesses/sec: ~143 years

16-character password, lowercase only (26 chars)

Entropy = 16 × log₂(26) = 16 × 4.70 = 75.2 bits

Time to crack at 1 billion guesses/sec: ~1.2 billion years

4-word passphrase (7,776 word dictionary)

Entropy = 4 × log₂(7776) = 4 × 12.9 = 51.7 bits

Time to crack at 1 billion guesses/sec: ~71 years

Why Length Beats Complexity

The Mathematics of Length

Adding one character to your password increases entropy more than adding character types:

Password Type Example Entropy Crack Time*
8 chars, complex P@ssw0rd! 52.6 bits 143 years
12 chars, simple correcthorse 56.4 bits 2,283 years
16 chars, simple correcthorsebattery 75.2 bits 1.2 billion years
20 chars, simple correcthorsebatterystaple 94.0 bits 627 trillion years

*At 1 billion guesses per second

The Exponential Effect

Each additional character multiplies the possible combinations by the character set size. This exponential growth quickly outpaces the linear benefit of expanding the character set.

Real vs. Theoretical Entropy

Entropy Killers

Theoretical entropy assumes perfect randomness. Real passwords often have much lower effective entropy due to:

  • Dictionary words: "Password1!" has low entropy despite mixed characters
  • Common substitutions: "@" for "a", "3" for "e", "0" for "o"
  • Keyboard patterns: "qwerty", "123456", "zxcvbn"
  • Personal information: Names, dates, addresses
  • Predictable patterns: Capital first letter, numbers/symbols at end

Effective Entropy Examples

Password Theoretical Entropy Effective Entropy Why Lower?
P@ssw0rd! 59 bits ~20 bits Common word with substitutions
John1985! 59 bits ~15 bits Name + birth year pattern
qwerty123 53 bits ~10 bits Keyboard pattern
Tr0ub4dor&3 72 bits ~28 bits Common substitution pattern

Entropy Requirements by Use Case

Minimum Entropy Recommendations

Use Case Minimum Bits Example Password Rationale
Low-value accounts 40-50 8-10 random chars Deters casual attacks
Email/Social Media 60-70 12-14 random chars Protects personal data
Financial accounts 70-80 14-16 random chars High-value target
Cryptocurrency 80-100 16-20 random chars Irreversible transactions
Encryption keys 128+ 24+ random chars Long-term security

Future-Proofing Your Passwords

Computing power doubles approximately every two years (Moore's Law). A password that takes 100 years to crack today might take only 50 years in two years, 25 years in four years, and so on.

Recommended minimums for 20-year security:

  • Add 10 bits to current requirements
  • Use at least 80 bits for important accounts
  • Consider 100+ bits for critical data
  • Implement 2FA regardless of password strength

Passphrases: A Different Approach

Diceware Method

Diceware uses a list of 7,776 words (6^5), each word adding 12.9 bits of entropy:

Words Entropy Example Crack Time*
3 38.8 bits correct horse battery 8.7 years
4 51.7 bits correct horse battery staple 71 years
5 64.6 bits correct horse battery staple cloud 584,542 years
6 77.5 bits correct horse battery staple cloud mountain 4.8 billion years

*At 1 billion guesses per second

Passphrase Advantages

  • Memorable: Stories and sentences stick in memory
  • High entropy: Long length compensates for predictability
  • Easy to type: No special characters or shift keys
  • Language-independent: Works with any language's word list

Attack Speed Considerations

Realistic Attack Speeds

Attack Type Speed (guesses/sec) Scenario
Online attack 10-1,000 Website login with rate limiting
Offline (slow hash) 10,000-100,000 bcrypt, scrypt, Argon2
Offline (fast hash) 1-100 billion MD5, SHA-1 (insecure)
GPU cluster 100 billion - 1 trillion Professional crackers
Nation-state Trillions+ Classified capabilities

Time to Crack by Entropy

At 100 billion guesses per second (high-end GPU cluster):

  • 40 bits: 11 seconds
  • 50 bits: 3 hours
  • 60 bits: 133 days
  • 70 bits: 374 years
  • 80 bits: 383,000 years
  • 90 bits: 392 million years
  • 100 bits: 401 billion years

Improving Your Password Entropy

Quick Wins

  1. Add length: Each character exponentially increases possibilities
  2. True randomness: Use a password generator, not patterns
  3. Unique passwords: Never reuse, even with variations
  4. Avoid predictability: No personal info or common substitutions
  5. Consider passphrases: Long and memorable beats short and complex (learn more in our guide on why passphrases beat traditional passwords)

Entropy Checklist

Does your password have:

  • ☐ At least 60 bits of entropy for important accounts?
  • ☐ True randomness (not keyboard patterns)?
  • ☐ No dictionary words or names?
  • ☐ No predictable substitutions (@ for a)?
  • ☐ Sufficient length (12+ characters)?
  • ☐ No connection to personal information?
  • ☐ Uniqueness (never used elsewhere)?

Entropy in Practice

Creating High-Entropy Passwords

  • Random password generators: Ensure cryptographic randomness
  • Diceware passphrases: Roll physical dice for true randomness
  • Password managers: Generate and store high-entropy passwords
  • Hardware RNG: Some devices use atmospheric noise or radioactive decay

Testing Password Entropy

Be cautious about online password strength checkers. Never enter your actual password. Instead:

  • Use similar patterns with different characters
  • Calculate entropy manually using the formula
  • Use offline tools that don't transmit data
  • Focus on method rather than specific passwords

Key Takeaways

  • Entropy measures password unpredictability in bits
  • Length contributes more to entropy than complexity
  • Aim for 60+ bits for important accounts, 80+ for critical
  • Real entropy is often lower than theoretical due to patterns
  • Passphrases can achieve high entropy while remaining memorable
  • Computing advances require planning for future attack speeds
  • True randomness is essential—avoid patterns and personal info
  • Combine high entropy with 2FA for maximum security (see our complete guide to two-factor authentication)