Password Entropy Explained (With Simple Examples)

Introduction

Password entropy is one of the most important concepts in password security, yet it's often misunderstood. In this guide, we'll break down entropy in simple terms, show you how to calculate it, and explain why it matters more than complexity alone.

What Is Password Entropy?

Entropy measures the unpredictability of your password. The higher the entropy, the more difficult it is for an attacker to guess or crack your password through brute force attacks.

Simple Definition

Think of entropy as the number of "yes/no questions" an attacker would need to ask to guess your password:

• Low entropy (40 bits): ~1 trillion guesses
• Medium entropy (60 bits): ~1 quintillion guesses
• High entropy (80 bits): ~1 septillion guesses
• Excellent entropy (100+ bits): Effectively unguessable

The Entropy Formula

Entropy is calculated using this formula:

Entropy (bits) = log₂(possible_combinations)

Or more practically:

Entropy = length × log₂(charset_size)

Example Calculation

For a 16-character password using all character types (94 possible characters):

Charset size: 94 (26 uppercase + 26 lowercase + 10 numbers + 32 symbols)
Length: 16 characters

Entropy = 16 × log₂(94)
Entropy = 16 × 6.55
Entropy ≈ 105 bits

This means there are 2^105 (about 40 undecillion) possible combinations.

Entropy by Character Set

Different character sets provide different amounts of entropy per character:

| Character Set | Size | Bits per Char | 16-Char Entropy | |--------------|------|---------------|-----------------| | Lowercase only | 26 | 4.7 | 75 bits | | Lower + Upper | 52 | 5.7 | 91 bits | | Alphanumeric | 62 | 5.95 | 95 bits | | All printable | 94 | 6.55 | 105 bits |

Key insight: Adding symbols increases entropy by ~10 bits for a 16-character password.

Entropy vs. Length

Length has a multiplicative effect on entropy:

| Length | Lowercase Only | All Characters | |--------|---------------|----------------| | 8 chars | 38 bits | 52 bits | | 12 chars | 56 bits | 79 bits | | 16 chars | 75 bits | 105 bits | | 20 chars | 94 bits | 131 bits | | 24 chars | 113 bits | 157 bits | | 32 chars | 150 bits | 210 bits |

Notice how length matters more than complexity for achieving high entropy.

Real-World Examples

Low Entropy (Weak)

Password: password123

• Length: 11 characters
• Charset: lowercase + numbers (36)
• Entropy: 11 × log₂(36) ≈ 57 bits

Despite being 11 characters, this password is weak because:

• It's a dictionary word
• It follows a common pattern
• Effective entropy is much lower (~20 bits)

Medium Entropy (Fair)

Password: MyDog2024!

• Length: 10 characters
• Charset: mixed (94)
• Theoretical entropy: 10 × 6.55 ≈ 66 bits

Better, but still vulnerable because:

• Contains dictionary words
• Follows predictable pattern (Word + Year + Symbol)
• Effective entropy: ~40 bits

High Entropy (Strong)

Password: K9#mL2$pQ7@nR4!v

• Length: 16 characters
• Charset: truly random (94)
• Entropy: 16 × 6.55 ≈ 105 bits

Excellent because:

• Completely random
• No patterns
• Full character set
• Generated by our random password creator

Why Patterns Reduce Entropy

Even if you use all character types, patterns dramatically reduce effective entropy.

Pattern Examples

Pattern: Aa1!Bb2@Cc3#Dd4$

• Theoretical entropy: 16 × 6.55 = 105 bits
• Effective entropy: ~30 bits (pattern is obvious)

Pattern: January2024!@#$

• Theoretical entropy: 15 × 6.55 = 98 bits
• Effective entropy: ~25 bits (dictionary word + year + sequence)

Learn more about avoiding patterns.

Entropy and Attack Resistance

Brute Force Attacks

Time to crack based on entropy (assuming 1 billion guesses/second):

| Entropy | Combinations | Time to Crack | |---------|-------------|---------------| | 40 bits | 1 trillion | 17 minutes | | 50 bits | 1 quadrillion | 12 days | | 60 bits | 1 quintillion | 36 years | | 70 bits | 1 sextillion | 37,000 years | | 80 bits | 1 septillion | 38 million years | | 100 bits | 1 nonillion | 40 billion years | | 105 bits | - | 1.3 trillion years |

A 16-character random password with 105 bits is effectively uncrackable.

Dictionary Attacks

Dictionary attacks bypass entropy by trying common words and patterns:

• "password123": Cracked instantly (in dictionary)
• "correct-horse-battery-staple": Minutes to hours (word list)
• "K9#mL2$pQ7@nR4!v": Trillions of years (no shortcuts)

This is why true randomness is critical.

Calculating Entropy for Different Password Types

Random Passwords

For truly random passwords, use the standard formula:

Entropy = length × log₂(charset_size)

Use our Strong Password Generator to create random passwords with known entropy.

Passphrases

For passphrases, entropy depends on word list size:

Entropy = number_of_words × log₂(word_list_size)

Example: 4 words from a 7,776-word list (Diceware):

Entropy = 4 × log₂(7776) ≈ 51 bits

Compare this to random passwords vs passphrases.

Hybrid Passwords

For passwords with patterns, effective entropy is much lower than theoretical entropy. Always assume the worst case.

Entropy Requirements by Use Case

Minimum Recommendations

| Account Type | Minimum Entropy | Recommended Length | |-------------|-----------------|-------------------| | Low-risk accounts | 60 bits | 12 characters | | Standard accounts | 80 bits | 16 characters | | Important accounts | 100 bits | 20 characters | | Critical systems | 120+ bits | 24-32 characters |

Industry Standards

• NIST: Recommends 80+ bits for user-chosen passwords
• OWASP: Recommends 128+ bits for generated passwords
• NSA: Requires 128+ bits for classified systems

How to Maximize Entropy

1. Use All Character Types

Each additional character type increases entropy:

Lowercase only:     26 characters → 4.7 bits/char
+ Uppercase:        52 characters → 5.7 bits/char
+ Numbers:          62 characters → 5.95 bits/char
+ Symbols:          94 characters → 6.55 bits/char

2. Increase Length

Length has the biggest impact:

• Adding 4 characters adds ~26 bits (94-char set)
• Doubling length doubles entropy
• Length beats complexity

3. Use True Randomness

• Generated passwords: Full theoretical entropy
• Patterned passwords: Reduced effective entropy
• Dictionary passwords: Minimal effective entropy

4. Avoid Predictable Patterns

These patterns reduce entropy:

• Dictionary words
• Keyboard walks (qwerty, asdfgh)
• Repeated characters (aaa, 111)
• Sequential patterns (abc, 123)
• Personal information (names, dates)

Entropy in Password Managers

Good password managers display entropy for generated passwords:

• 1Password: Shows "Strength" based on entropy
• Bitwarden: Displays entropy in bits
• LastPass: Shows strength meter

Our Strong Password Generator shows both entropy and strength rating.

Common Misconceptions

Myth 1: "Complexity = Security"

False. A 12-character random lowercase password (56 bits) is stronger than "P@ssw0rd!" (40 bits effective).

Myth 2: "Longer is always better"

Mostly true, but only if the password is random. A 30-character dictionary phrase may have less entropy than a 16-character random password.

Myth 3: "Substitutions add security"

False. "P@ssw0rd" is barely better than "Password" because these substitutions are well-known to attackers.

Myth 4: "I need to memorize high-entropy passwords"

False. Use a password manager to store high-entropy passwords. Only memorize your master password.

Practical Entropy Guidelines

For Manual Passwords

If you must create a memorable password:

1. Use at least 4 random words (Diceware method)
2. Add random numbers and symbols between words
3. Aim for 70+ bits of entropy
4. Consider passphrases vs random passwords

For Generated Passwords

Use our Strong Password Generator with:

1. 16+ characters for standard accounts (105+ bits)
2. 20+ characters for important accounts (131+ bits)
3. All character types enabled
4. True randomness (no patterns)

Entropy and Future-Proofing

Moore's Law

Computing power doubles roughly every 18-24 months. This means:

• 60-bit password today → 58-bit equivalent in 2 years
• 80-bit password today → 78-bit equivalent in 2 years
• 100-bit password today → 98-bit equivalent in 2 years

Solution: Use 100+ bits to stay ahead of computing advances.

Quantum Computing

Quantum computers could theoretically halve effective entropy:

• 100-bit password → 50-bit effective (still strong)
• 128-bit password → 64-bit effective (excellent)
• 200-bit password → 100-bit effective (maximum)

32-character passwords provide quantum resistance.

Testing Your Password's Entropy

Online Tools

Use our Strong Password Generator to see:

• Real-time entropy calculation
• Strength rating (Weak/Fair/Good/Strong/Excellent)
• Estimated crack time

Manual Calculation

1. Count the character set size
2. Measure the password length
3. Apply the formula: length × log₂(charset_size)
4. Compare to recommended minimums

Conclusion

Password entropy is the ultimate measure of password strength:

✅ Aim for 100+ bits for excellent security
✅ Use random generation for maximum entropy
✅ Prioritize length over complexity
✅ Avoid patterns that reduce effective entropy
✅ Use a password manager to handle high-entropy passwords

Ready to create a high-entropy password? Use our Strong Password Generator to generate passwords with 105+ bits of entropy instantly.

Related Reading

• Random Password Creator: How True Randomness Improves Security
• Password Length vs Complexity: Which Matters More?
• How Hackers Crack Weak Passwords