1) Determine the multiplicative inverse of x^3+x+1 in GF(16) with m(x) = x^4+x+1. (30 poitnts)

2) compute the greatest common divisor of x^8 + x^5 + x^4 + x + 1 and x^7 + x^6 + x^5 + x + 1, as polynomials with coefficients in F2 = GF(2). (30 points)

3) (40 points) Given the plain text "AES uses poly math", and the key {010101010101010101010101010101}:

a. show the original contents of State, displayed as a 4x4 matrix. (you can use ASCII table and use z for padding if needed).

b. Show the value of state after SubBytes.

c. Show the value of state after ShiftRows (performed after SubBytes).

d. Show the value after state after MixColumns (performed after ShiftRows). (you can show only one element of the matrix for this one!)