$(5)(6$ pts$)$ Consider a sequence generator implemented with the shift register where the value of

bit i will be sent to bit $($i$+1)$ after a clock cycle. But the value of bit $0$ is the result of XOR $($bit $2,$

bit $4).$

The state of the sequence generator is $b0,b1,b2,b3$ and $b4.$ The initial state of the sequence

generator can be anything except state $00000.$ The sequence value, taken from bit $4,$ will be

used to encrypt a data stream bit by bit. The total length of the sequence is $31.$ That is$,$ after $31$

clock cycles, the sequence taken from bit $4$ will repeat itself.

The initial state $($i$.$e$.$ state at $t=0)$ is determined by the following equation:

Initial state $=$ IV $($initial vector$)$ xORKV$(keyvariable),$

where IV and KV are two $5-$bit vectors. IV$,$ selected by the transmitter, will be transmitted along

with the encrypted data. $KV$ is the shared key between the sender and the receiver. $KV,$ of

course, cannot be revealed; otherwise, the cipher will be broken.

$($a$)$ Why do we need IV$?$

$($b$)$ Write down the entire sequence, of length $31,$ of this generator $($starting from the state $=$

$.$

$($c$)$ Assume at $t=0,IV=[10101]$ and $KV=[10111],$ and the plain text is

$10111101($the leftmost bit is generated at $t=1).$

Write down the cipher text.