1. Let A be a non-singular matrix over GF(2). Consider the affine transformation f(b) = A b + c, where c 0 Determine the inverse transformation. Is this also an affine transformation?
2. The following matrix over GF(2) is used in the SubBytes Transformation in AES. Find its inverse.

1 0 0 0 1 1 1 1

1 1 0 0 0 1 1 1

1 1 1 0 0 0 1 1

1 1 1 1 0 0 0 1

1 1 1 1 1 0 0 0

0 1 1 1 1 1 0 0

0 0 1 1 1 1 1 0

0 0 0 1 1 1 1 1

4. Let Z₂[x] represent all finite degree polynomials with coefficients in GF(2). Construct the finite field Z₂[x]/( x⁴+x+1 ) which is isomorphic to GF(2⁴). Let be a root of the primitive polynomial x⁴+x+1. Develop a table to show the relationship between the multiplicative and additive representation of each element of this finite field.