[Galois Field] We’ll do some arithmetic in GF(16), which is generated by the primitive polynomial pX=x4+x+1. Below is th

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[Galois Field]
We’ll do some arithmetic in GF(16), which is generated by
the primitive polynomial pX=x4+x+1. Below is the table of elements
from class.
Name
Polynomial
Vector
Binary
0
0
(0,0,0,0)
0000
1
1
(0,0,0,1)
0001
a
x
(0,0,1,0)
0010
b
x+1
(0,0,1,1)
0011
c
x2
(0,1,0,0)
0100
d
x2+1
(0,1,0,1)
0101
e
x2+x
(0,1,1,0)
0110
f
x2+x+1
(0,1,1,1)
0111
g
x3
(1,0,0,0)
1000
h
x3+1
(1,0,0,1)
1001
i
x3+x
(1,0,1,0)
1010
j
x3+x+1
(1,0,1,1)
1011
k
x3+x2
(1,1,0,0)
1100
l
x3+x2+1
(1,1,0,1)
1101
m
x3+x2+x
(1,1,1,0)
1110
n
x3+x2+x+1
(1,1,1,1)
1111
a) Calculate the sum m+i. Show your work.
b) What is the additive inverse of j? Note: find an
element α∈GF(16) so that
j+α=0.
c) Calculate the product d⋅e. Show your work. Note: name
the resulting element, so be sure to reduce modulo
x4+x+1.