Password Security

Probability · Axiom Academy

How permutations and combinations protect (or fail to protect) your digital life Every day, you protect your email, bank account, social media, and more with passwords. But have you ever wondered: what makes a password truly secure? The mathematics of password security isn't about memorizing complex rules. It's about understanding the size of the search space —how many possible passwords an attacker would need to try to crack yours. Let's explore how probability theory determines whether your password is a steel vault or a cardboard box. Let's build a password and calculate how many possible combinations exist. Choose your character types: Why does the password space grow so quickly? It's all about permutations with repetition . n = number of possible characters (your alphabet size) For each position in your password, you can choose any character from your alphabet—that's what makes it a permutation with repetition. Order matters (abc ≠ bca), and you can reuse characters (aaa is valid). Let's see how long it would take an attacker to crack passwords of different strengths using brute force (trying every combination). Assumptions: Attacker can try 1 billion passwords per second (realistic with modern GPU) Here's a common debate in password security: Which password is more secure against brute force attacks? The Length vs. Complexity Trade-off The surprising truth: length matters more than complexity for password security!

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