Mathematically, how long would it take to crack a bcrypt password hash?

Question

So I'm currently using bcrypt to hash passwords with a randomly generated salt (as seen in the pip bcrypt module), with 12 rounds.

I have been looking around, but I cannot find a detailed and clear mathematical way to estimate how long it would take for a strong GPU (or even other methods of cracking) to crack my bcrypt hash.

I found this gist which descibes 8x Nvidia GTX 1080 Hashcat Benchmarks, but I can't seem to use the figures provided for bcrypt to apply to my implementation of the hashing algorithm.

Answer 1

Ok, so from the article you linked, we learn that the 8x Nvidia setup can calculate about 100 000 bcrypt hashes per second (H/s). The cost factor used for the bench mark is 5 - very low - if the comments are to be believed. For bcrypt, the number of rounds equals two to the power of the cost factor. So if yours is 12, your hash will be 2¹²/2⁵ = 2⁷ = 128 times slower. So from 10⁵ H/s you're down to 10³ H/s.

Now all you need to know is that to crack a hash with n bits of entropy, on average you need to try 2^n-1 times. So take a password consisting of 8 random lower case letters for instance. It has an entropy of n = log₂(26⁸) = 38 bits. To crack it you would need 2^38-1/1000 seconds = 4 years.

Note that the benchmark is from 2016. As time passes by, hardware gets faster. You will need to regularly reevaluate your cost factor to stay up to date.

Answer 2

This is a common misconception about passwords. It depends on your definition of password. You have to first define what you mean by password, such as "8 alphanumeric characters". Once you define that, then we can calculate.

An "alphanumeric" is 26 upper case, 26 lower case, and 10 numeric characters, or 62 possible combinations.

To try all combinations of an 8 character password, that would be: 62 * 62 * 62 * 62 * 62 * 62 * 62 * 62

That's 218,000,000,000,000.

Divided this by 13094 (the number of hashes per second), and you get 16674820955 seconds, or 528 years.

The impracticality of trying all combinations leads to other strategies, such as instead choosing dictionary words or "Markov Chains" to test the sorts of passwords are likely to choose, instead of trying all random combinations. You can't mathematically calculate how fast this is because it's entirely subjective. It depends upon luck, what passwords the targets have chosen, and what passwords you've chosen to test. It's often something of the order of a 10% chance of cracking a password with a few hours of cracking, but again, that's subjective experience. Depending on whose password you are cracking and how you are cracking it, your experience will vary.

Anyway, the point is that this question has no answer. It's a misconception of how things work. And that's why you can't google the answer.