multiplicative inverse

Question

I actually do want to know how to generate the multiplication inverse (M.I) table, I mean the calculation how the M.I been generated.. such as the M.I of 22 is 5A regarding to the table.. Do you know how to calculate M.I how the 22 will become 5A

Answer 1

22 is the hex encoding of a binary polynomial $f = x^5 + x$ that is used to represent the element of the AES finite field (the field is defined as $F_{2^8} = GF(2)[x] / p(x)$) where $p(x) = x^8+x^4+x^3+x+1$ is the reduction polynomial.

Now, we need to find the multiplicative inverse of $f(x)$, i.e. another polynomial $g(x)$ such that $$ f(x)\cdot g(x) \equiv 1 \pmod{p(x)} $$ We can do this either by using extended Euclid algorithm for polynomials or using the fact that $f^{-1}(x) = f^{254}(x)$ because of the Little Fermat's Theorem.

Using a computational algebra package like e.g. GP/PARI we can compute

p = Mod(1,2)*x^8 + Mod(1,2)*x^4 + Mod(1,2)*x^3 + Mod(1,2)*x + Mod(1,2)
f = Mod(1,2)*x^5 + Mod(1,2)*x
g = Mod(f,p)^-1

and we get the result

Mod(Mod(1, 2)*x^6 + Mod(1, 2)*x^4 + Mod(1, 2)*x^3 + Mod(1, 2)*x, Mod(1, 2)*x^8 + Mod(1, 2)*x^4 + Mod(1, 2)*x^3 + Mod(1, 2)*x + Mod(1, 2))

so we get the reduced polynomial $x^6 + x^4 + x^3 + x$ and this encodes as 5A, the table entry that corresponds to 22.

The only element that does not have its multiplicative inverse is 00, so by convention we define that 00 is mapped to 00 again.

Answer 2

As pointed in Krystian's comment, a former version of this answer was wrong. See correction at end of that other answer to the similar question that justified closing the present question.