Question

5. A hacker is trying to guess a password. They have worked out that the password has length 8, being a mix of 1 uppercase letter (from {A. . . . ,7), 5 lowercase letters (from {a, . . . ,:) and 2 digits (from 10,...,91). They do not know in which order these symbols occur in the password. [In this question, answers may be given as a product of integers, without evaluating further.] a) How many passwords fit this description? (b) If they find out that all 8 symbols in the password are different, how many passwords are possible? (c) Suppose that all symbols in the password are different, and the hacker discovers the position of the lowercase letters and that one of them is 'w' (without knowing which one). How many passwords are possible? (d) Suppose we are the situation described in part (c). The hacker writes a program that repeatedly chooses one of the possible passwords uniformly at random, and tries it. The program tries six thousand such trials per second. What is the expected value of the number of minutes until the hacker cracks the password

Answer 1

a.

In the position where we have a uppercase letter, there are 26 different ways to fill this position. Further for positions where we have lower case letters, again there are 26 ways to fill them. We have 5 such positions to fill. So total ways to fill this 5 spots will be 26x26x26x26x26=26^5. Lastly for positions where we have digits from 0 to 9, there will be 10 ways to fill them. We have 2 such positions to fill, which can be done in 10x10=100 ways.

But we have 8 different spots and we can arrange the available characters in 8! ways. This is to account for the fact that any digit character (0 to 9) any uppercase character or any lower case character, can take any of the available 8 positions.

So, the total possible passwords will be,

(26)(26^5)(100)(8!)

b.

If all 8 symbols in the password are to be different , that means if we have say "a" at one position, we can't have it again in any other position, although we will have 4 more positions to fill with lower case letters. Similarly if we have "0" in one position we can't have it in any other position, although we will have 1 more position to fill with a digit. (As we only have one spot to fill with uppercase letters we don't need to consider any change in number of ways for this)

So the total passwords possible will be,

(26)(26x25x24x23x22)(10x9)(8!)

c.

If one position is known then we have only 7! ways left to arrange characters in 7 remaining positions. Also, we know that one lower case character is "w" which leaves us with 25 lower case characters to fill 4 more positions in which there will be a lower case character. i.e. 25x24x23x22. There will be no change in the number of ways for arrangement of uppercase characters and digits

So, the number of ways will be,

(26)(25x24x23x22)(10x9)(7!)

d.

Considering that the program has to go through all the possible passwords before it cracks the correct one, then at a speed of 6000 per sec , the total time required to crack the password will be,

t=(26)(25x24x23x22)(10x9)(7!)/(6000) = 596,756,130 secs

In minutes it will be,

(596,756,130)/60 = 9,945,936 mins