Suppose a password based authentication system requires the password to be $15$ char$-$

acters long and it should contain exactly two digits, exactly two of the six special

characters, namely, $\{!,$@$,$#$,$$$,\%,$&$\}$ and the rest of the characters can be upper and

lower case letters. How many such passwords are possible?

$($a$)$ Assume that the password file contains encrypted passwords and the password

file is available for the attacker. If the attacker has access to a supercomputer

which can encrypt ${10}^{15}$ passwords per second, how many years will it take for the

attacker to determine the password of a user using brute force attack?

$($b$)$ If the password file is salted with a salt of length $128$ bits and the slow hash

function used for computing the hash of one password takes $5$ seconds on a single

node of a super computer. What is the minimum number of years an attacker will

take to compute the rainbow table of all possible passwords for such a system, if

the attacker has access to a super computer containing $10$ million such nodes?$(16$

points$)$