Calculation of 0x84

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x84 =  10000100 = x^7 + x^2

m(x) = (x) * a(x) + (x^4 + x + 1)

Calculation of 0x84^-1 in the finite field GF(2⁸)

00010011 = 00000001 * 100011011 + 00000010 * 10000100
00001111 = 00001001 * 100011011 + 00010011 * 10000100
00000010 = 00011010 * 100011011 + 00110111 * 10000100
00000001 = 01001111 * 100011011 + 10010110 * 10000100
00000000 = 10000100 * 100011011 + 100011011 * 10000100

a^-1(x) = x^7 + x^4 + x^2 + x = 10010110 = 0x96

The calculation of 0x84^-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      0     1     1
           1 1 0 0 0 1 1 1      1     1     1
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(84) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      0     1     0
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(84) = 01011111 = 5f

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com