Password Math

Math of Password Cracking

Suppose passwords are 8 characters, where a character can be an upper-case letter, lower-case letter, digit or special (printable) character. If there are 32 special characters, then 94⁸ > 2⁵² different passwords are possible.
Suppose passwords are hashed. Also, suppose there is a dictionary of 2²⁰ passwords and we estimate that a randomly selected password will be in the dictionary with probability 1/2.
- If we want to crack a single password then
  1. Without using the dictionary, the expected work is about 2⁵¹
  2. With the dictionary, the expected work is about
    
    1/2 2⁵¹ + 1/2 2¹⁹ ≈ 2⁵⁰
- If we have a file of 128 password hashes and we would like to crack any one of these, then
  1. Without using the dictionary, the expected work is about 2⁴⁴ assuming that no salt is used. If the password hashes are salted, then the work is about 2⁵¹
  2. With the dictionary, the probability that at least one of the 128 passwords is in the dictionary is
    
    1 - (1/2)¹²⁸ ≈ 1
    
    So we can neglect the case where none of the passwords are in the dictionary. Using the dictionary on this password file, the expected work factor is
    
    1/2 2¹⁹ + 1/2² (2²⁰ + 2¹⁹) + 1/2³ (2 2²⁰ + 2¹⁹) + ⋅ ⋅ ⋅ + 1/2¹²⁸ (127 2²⁰ + 2¹⁹) = 6 2¹⁸ < 2²¹
    
    assuming that the hashes are salted. If the hashes are not salted, we could precompute the hashes of the entire dictionary (2²⁰ hashes) and amortize this work factor over the number of times that the attack is used.