This problem concerns binary representations. Complete the follow$-$

ing, in each case describing how you arrive at your answer.

$($a$)$ AIlow many integers can be represented using nine binary digits?

$($b$)$ What is the smallest nine$-$bit integer? What is the largest nine$-$

bit integer?

$($c$)$ What is the minimum number of binary digits needed to represent

the integer $250000?($note: you can answer this without finding the full

binary represientation of $250000)$

$($a$)$ Given that the ciphertext EBO was encrypted using a shift$($n$)$

cipher and that the plaintext is an English word, if you use a brute

force key search to try to decrypt it$,$ you will run into an ambiguity.

Describe what the issue is

$($b$)$ Is the issue in part $($a$)$ resolved if you find out that the plaintext

is a proper first name rather than a word? Explain.

$($c$)$ Compare and contrast the decryption of this message with the

decryption you did in the Module $1$ discussion. For example, you might

focus on the process you used for each, the difficulties encountered, etc.

In our lecture notes we used the initial state $s_0{s}_1{s}_2=011$ and

recursion

$s_m-=s_{m-1}+s_{m-3},(mod2)$

to extend this key stream to $011101.$

$($a$)$ Continue to extend this stream until you see a repeating pattern.

What block of digits is continually repeated?

$($b$)$ Describe how you can use what you found in part $($a$)$ to predict

the key stream that would result from starting with the initial state

$s_0{s}_1{s}_2-010.$

$($c$)$ Repeat $($a$)$ with the recursion

$s_m-=s_{m}1+s_m-4,(mod2)$

and initial state $s_0{s}_1{s}_2{s}_3=1000.$

Suppose you are using a variable shift cipher shift $(n_1,n_2,$dots,$n_l)$ to

encrypt messages and you wish to thwart Oscar, who has the following

capabilities: $(1)$ he can figure out the value of $l$ that you are using, and

$(2)$ he has the capacity to perform a brute force key search over one

billion keys.

To give yourself a measure of security, you would like Oscar$$s ex$-$

pected average search length to be at least one trillion trials. What is

the minimum value of l required? Explain carefully how you arrive at

your answer.