The MixColumns transformation of AES consists of a matrix-vector multiplication in the field with an irreducible polynomial . Let b = (b₇x⁷+. . .+b₀) be one of the (four) input bytes to the vector-matrix multiplication. Each input byte is multiplied with the constants 01, 02, and 03. Your task is to provide exact equations for computing those three constant multiplications after reductions. We denote the result by d = (d₇x⁷+. . .+d₀).

1. Equations for computing the 8 bits of d = 01 b mod P(x).

2. Equations for computing the 8 bits of d = 02 b mod P(x).

3. Equations for computing the 8 bits of d = 03 b mod P(x).

Note: The AES specification uses 01 to represent the polynomial 1, 02 to represent the polynomial x, and 03 to represent x+1.

We recall from the discussion of stream ciphers that a 2-input XOR gate performs a GF(2) addition.

How many 2-input XOR gates are required to perform one constant multiplication by 01, 02, and 03, respectively, in GF(2^8)?

GF(28) PI) = x +2 +2 +2+1