Increasing costs impact some players more than others. Those who know about power laws and know how to prioritize are impacted less than those who naively believe everything is equally important.

This post will look at economics and power laws in the context of password cracking. Increasing the cost of verifying a password does not impact attackers proportionately because the attackers have a power law on their side.

## Key stretching

In an earlier post I explained how key stretching increases the time required to verify a password. The idea is that if authentication systems can afford to spend more computation time verifying a password, they can force attackers to spend more time as well.

The stretching algorithm increases the time required to test a single salt and password combination by a factor of *r* where *r* is a parameter in the algorithm. But increasing the amount of work per instance by a factor of *r* does not decrease the number of users an attacker can compromise by a factor of *r*.

## Power laws

If an attacker is going through a list of compromised passwords, the number of user passwords recovered for a fixed amount of computing power decreases by less than *r* because passwords follow a power law distribution [1]. The *n*th most common password appears with frequency proportional to *n*^{−s} for some constant *s*. Studies suggest *s* is between 0.5 and 0.9. Let’s say *s* = 0.7.

Suppose an attacker has a sorted list of common passwords. He starts with the most common password and tries it for each user. Then he tries the next most common password, and keeps going until he runs out of resources. If you increase the difficulty of testing each password by *r*, the attacker will have to stop after trying fewer passwords.

## Example

Suppose the attacker had time to test the 1000 most common passwords, but then you set *r* to 1000, increasing the resources required for each individual test by 1000. Now the attacker can only test the most common password. The most common password accounts for about 1/24 of the users of the 1000 most common passwords. So the number of **passwords** the attacker can test went down by 1000, but the number of **users** effected only went down by 24.

It would take 1000 times longer if the attacker still tested the same number of passwords. But the attacker is hitting diminishing return after the first password tested. By testing passwords in order of popularity, his return on resources does not need to go down by a factor of 1000. In this example it only went down by a factor of 24.

How much the hacker’s ROI decreases depends on several factors. It never decreases by more than *r*, and often it decreases less.

## Related posts

[1] Any mention of power laws brings out two opposite reactions, one enthusiastic and one pedantic. The naive reaction is “Cool! Another example of how power laws are everywhere!” The pedantic reaction is “Few things actually follow a power law. When people claim that data follow a power law distribution the data usually follow a slightly different distribution.”

The pedantic reaction is technically correct but not very helpful. The enthusiastic reaction is correct with a little nuance: many things approximately follow a power law distribution over a significant range. Nothing follows a power law distribution exactly and everywhere, but the same is true of any probability distribution.

It doesn’t matter for this post whether data exactly follow a power law. This post is just a sort of back-of-the-envelope calculation. The power law distribution has some empirical basis in this context, and it’s a convenient form for calculations.