Question $2(11$ points$)$:

Assume an attacker knows that a user's password is either $"$wxyz$"$ or $"$bddf$".$ Say the user encrypts his password using:

a$)$ The substitution cipher,

b$)$ The Vigen$$re cipher using period $2,$ or

c$)$ The Vigen$$re cipher using period $3$

and the attacker sees the resulting ciphertext. Show how the attacker can determine the user's password, or explain why this is not possible.

Question $3(6$ points$)$:

Suppose we have a computer with a $4\mathrm{.}2GHz16-$core processor that executes $4\mathrm{.}2{10}^9$ cycles per second per core. Considering that it can test $20$ keys per CPU cycle, calculate:

a$)$ The average time $($in years$)$ to find a key by the brute$-$force attack if the key size is $56$ bits.

b$)$ The average time $($in years$)$ to find a key by the brute$-$force attack if the key size is $128$ bits

Question $4(16$ points$)$:

Answer each of the following without using a calculator.

a$)3-11(mod9)=$

b$)1529(mod13)=$

c$)-\frac{12}{35}(mod19)=$

d$)$ Are $172$ and $68$ co$-$prime numbers?