< S(A; B, C) for any systems A, B, C. According to the definition of mutual information, we have S(A;B)=S(A)+S(B)-S(A,B) = and S(A; B, C) = S(A) + S(B, C) - S(A, B, C), which yields by substitution S(A, B, C) + S(B) < S(A, B) +S(B,C). This last prop- erty is referred to as strong subadditivity inequality. Its demonstration requires no less than two additional theorems (including the so-called Lieb's theorem), both being rather mathematically involved! Therefore, here we shall take the "strong subadditivity" prop- erty as a given postulate, which closes the loop. Combining properties (i)-(iii) we finally obtain S(P; Q) > S'(P;R), which is the inequality in Eq. (X16). Appendix Y (Chapter 25) Polynomial byte representation and modular multiplication The novelty introduced by the AES cryptosystem is the use of modular multiplication by byte blocks, which is described in this appendix. For multiplied bytes to remain within a one-byte size, this multiplication must be defined modulo a certain "prime" number. This requires one to introduce another representation for binary numbers, which is the polynomial (byte) representation. A polynomial is an element that can be defined according to the expression m(x) = Cnx" + Cn-1xn-1 FCIX + co, where Cn, Cn-1 C1, CO are called the poly- nomial coefficients and n is the polynomial's degree. Since a byte is the eight-bit word C7C6C5C4C3C2C1C0 (Ck = 0 or 1), its equivalent polynomial representation is m(x) = C7X7 + C6X6... +cix+co. Let's look now at the XOR operation between two polynomials. Consider the two numbers {1010 0010} = {A3} and {0101 1111} = {5 F}. In binary, hexadecimal, and polynomial representations, we get {1010 0010} Q {0101 1111} = (1111 1101}, {A3} [5F} = {FD} The operation of polynomial multiplication, modulo a polynomial m(x), which we label with the symbol o, is no more difficult to grasp. The idea is that the reduction modulo m(x) should give a polynomial whose degree (here) is strictly less than eight, SO that the result of the multiplication can always be rep