Calculation of 0x9c

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x9c =  10011100 = x^7 + x^4 + x^3 + x^2

m(x) = (x) * a(x) + (x^5 + x + 1)

Calculation of 0x9c^-1 in the finite field GF(2⁸)

00100011 = 00000001 * 100011011 + 00000010 * 10011100
00010000 = 00000100 * 100011011 + 00001001 * 10011100
00000011 = 00001001 * 100011011 + 00010000 * 10011100
00000001 = 01110011 * 100011011 + 11111001 * 10011100
00000000 = 10011100 * 100011011 + 100011011 * 10011100

a^-1(x) = x^7 + x^6 + x^5 + x^4 + x^3 + 1 = 11111001 = 0xf9

The calculation of 0x9c^-1 is made with the Extended Euclidean algorithm. Instead of normal division and multiplication you need to use Polynomialdivision and Polynomialmultiplication.

Affine transformation over GF(2)

           1 0 0 0 1 1 1 1      1     1     0
           1 1 0 0 0 1 1 1      0     1     1
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      1     0     1
SBOX(9c) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      1     0     1

SBOX(9c) = 11011110 = de

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com