Exercise $2$: Lan Manager hash $[20$ pts$]$

Back in the days, up to Windows XP$,$ Windows account passwords were hashed using Microsoft's LAN

Manager $($LM$)$ hash function, which works as follows:

Step $1$ The password is converted into upper case, null$-$padded to $14$ characters $($or truncated to

$14$ characters$),$ and split into two $7-$character halves.

Step $2$ Each half is separately converted into a DES key. This key is used to encrypt the ASCII

string $"$KGS$!$@#$$\%",$ producing an $8-$byte value.

Step $3$ The two $8-$byte values are concatenated, resulting in a $16-$byte hash.

Suppose the attacker obtains a file with $N$ hashed passwords. How much work would he need to do$,$

at most, to crack these passwords by brute$-$force search? Show your calculations. Express the result

in scientific notation $),$ and round it to two decimals. Assume that

users could type any of the $95$ printable characters found on a US keyboard i$.$e$.,$ letters, numbers,

symbols, and punctuation marks, which are represented by codes $32$ to $126$ in the ASCII table.

Passwords could be any length.

$15pts$

Knowing that a single modern NVIDIA GeForce RTX $4090$GPU can perform $151\mathrm{.}1G\frac{H}{s}$ for LM$,{?}^2$

how long would it take an attacker with one such GPU to crack these $N$ hashes? Assume that the

overhead of matching a $64-$bit string in a list of $2N64-$bit strings is negligible.

$5pts$