Calculation of 0x91

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x91 =  10010001 = x^7 + x^4 + 1

m(x) = (x) * a(x) + (x^5 + x^4 + x^3 + 1)

Calculation of 0x91^-1 in the finite field GF(2⁸)

00111001 = 00000001 * 100011011 + 00000010 * 10010001
00000111 = 00000110 * 100011011 + 00001101 * 10010001
00000001 = 00110001 * 100011011 + 01101010 * 10010001
00000000 = 10010001 * 100011011 + 100011011 * 10010001

a^-1(x) = x^6 + x^5 + x^3 + x = 01101010 = 0x6a

The calculation of 0x91^-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     0
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      1     0     0
SBOX(91) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      0     0     1

SBOX(91) = 10000001 = 81

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com