$(1$ point$)$ This question concerns the field $GF(16).$ The modulus is $P(x)=x^4+x+1.(1$ point$)$ This question concerns the field $GF(256).$ The modulus is $P(x)=x^8+x^4+x^3+x+1.$

Please answer the following questions about arithmetic in this field.

a$)$ If $p(x)=x^7+x^5+x^4+x^2+x,$ what is the corresponding hexadecimal representation? Please prepend an $"$Ox$"$ on your answer and use lowercase letters $($if letters occur$).$

b$)$ If $p(x)=x^4,$ what is the corresponding hexadecimal representation? Please prepend an $"0$x$"$ on your answer and use lowercase letters $($if letters occur$).$

c$)$ Consider the hex byte $0xdb.$ What is the corresponding element of GF$(256)($as a polynomial$)?$

d$)$ Consider the hex byte $0x78.$ What is the corresponding element of $GF(256)($as a polynomial$)?$

Please answer the following questions about arithmetic in this field.

a$)$ If $p(x)=x^3+x$ and $q(x)=x^3+x^2,$ what is $p(x)+q(x)$ modulo $P?$

b$)$ If $p(x)=x^3+x$ and $q(x)=x^3+x+1,$ what is $p(x)+q(x)$ modulo $P?$

c$)$ If $p(x)=x^3+x^2$ and $q(x)=x+1,$ what is $p(x)*q(x)$ modulo $P?$

d$)$ If $p(x)=x^3$ and $q(x)=x^2+1,$ what is $p(x)*q(x)$ modulo $P?$