Calculation of 0xda

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xda =  11011010 = x^7 + x^6 + x^4 + x^3 + x

m(x) = (x + 1) * a(x) + (x^6 + x^5 + x^4 + x^2 + 1)

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

01110101 = 00000001 * 100011011 + 00000011 * 11011010
00110000 = 00000010 * 100011011 + 00000111 * 11011010
00010101 = 00000101 * 100011011 + 00001101 * 11011010
00001111 = 00001101 * 100011011 + 00010000 * 11011010
00000100 = 00010010 * 100011011 + 00111101 * 11011010
00000011 = 00111011 * 100011011 + 01010111 * 11011010
00000001 = 01011111 * 100011011 + 11000100 * 11011010
00000000 = 11011010 * 100011011 + 100011011 * 11011010

a^-1(x) = x^7 + x^6 + x^2 = 11000100 = 0xc4

The calculation of 0xda^-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      0     1     1
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      0     0     0
SBOX(da) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      0     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      1     0     0

SBOX(da) = 01010111 = 57

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com