Calculation of 0x86

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x86 =  10000110 = x^7 + x^2 + x

m(x) = (x) * a(x) + (x^4 + x^2 + x + 1)

Calculation of 0x86^-1 in the finite field GF(2⁸)

00010111 = 00000001 * 100011011 + 00000010 * 10000110
00000111 = 00001011 * 100011011 + 00010111 * 10000110
00000010 = 00110000 * 100011011 + 01100111 * 10000110
00000001 = 01011011 * 100011011 + 10111110 * 10000110
00000000 = 10000110 * 100011011 + 100011011 * 10000110

a^-1(x) = x^7 + x^5 + x^4 + x^3 + x^2 + x = 10111110 = 0xbe

The calculation of 0x86^-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     0
           1 1 0 0 0 1 1 1      1     1     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     0
SBOX(86) = 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 0      0     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(86) = 01000100 = 44

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com