$(1$ point$)$ Note: The notation from this problem is from Understanding Cryptography by Paar and Pelzl.

Consider the LFSR represented by the polynomial $x^4+x^2+x+1.$

What are the tap bits of the LFSR$?$ Please enter your answer as unspaced binary digits $($e$.$g$.010101$ to represent

$p_5=0,p_4=1,p_3=0,p_2=1,p_1=0,p_0=1.(1$ point$)$ Note: The notation from this problem is from Understanding Cryptography by Paar and Pelzl.

Suppose you have an LFSR with $6$ state bits. The first $12$ bits of output produced by this LFSR are

$001100011111=s_0{s}_1{s}_2{s}_3{s}_4{s}_5{s}_6{s}_7{s}_8{s}_9{s}_{10}{s}_{11}.$

The first bit produced is the leftmost bit and the bit most recently produced is the rightmost bit.

a$)$ What is the initial state of the LFSR$?$ Please enter your answer as unspaced binary digits $($e$.$g$.010101$ to represent

$\{$:$s_5=0,s_4=1,s_3=0,s_2=1,s_1=0,s_0=1).$

b$)$ What are the tap bits of the LFSR$?$ Please enter your answer as unspaced binary digits $($e$.$g$.010101$ to represent

$\{$:$p_5=0,p_4=1,p_3=0,p_2=1,p_1=0,p_0=1).(1$ point$)$ Note: The notation from this problem is from Understanding Cryptography by Paar and Pelzl.

Consider the LFSR represented by the polynomial $x^4+x^3+x^2+x+1.$

What are the tap bits of the LFSR$?$ Please enter your answer as unspaced binary digits $($e$.$g$.010101$ to represent

$p_5=0,p_4=1,p_3=0,p_7=1,p_1=0,p_0=1.(1$ point$)$ Note: The notation from this problem is from Understanding Cryptography by Paar and Pelzl.

We conduct a known$-$plaintext attack against an LFSR$.$ Through trial and error we have determined that the number of states is $m=4.$

The plaintext given by

$00110001=x_0{x}_1{x}_2{x}_3{x}_4{x}_5{x}_6{x}_7$

when encrypted by the LFSR produced the ciphertext

$01001000=y_0{y}_1{y}_2{y}_3{y}_3{y}_5{y}_6{y}_7.$

What are the tap bits of the LFSR$?$ Please enter your answer as unspaced binary digits $($e$.$g$.0101$ to represent $p_3=0,p_2=1,p_1=0,p_0=1).$