Calculation of 0xb5

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xb5 =  10110101 = x^7 + x^5 + x^4 + x^2 + 1

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

Calculation of 0xb5^-1 in the finite field GF(2⁸)

01110001 = 00000001 * 100011011 + 00000010 * 10110101
00100110 = 00000011 * 100011011 + 00000111 * 10110101
00011011 = 00000100 * 100011011 + 00001011 * 10110101
00001011 = 00001111 * 100011011 + 00011010 * 10110101
00000110 = 00010101 * 100011011 + 00100101 * 10110101
00000001 = 00110000 * 100011011 + 01110101 * 10110101
00000000 = 10110101 * 100011011 + 100011011 * 10110101

a^-1(x) = x^6 + x^5 + x^4 + x^2 + 1 = 01110101 = 0x75

The calculation of 0xb5^-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     1
           1 1 0 0 0 1 1 1      0     1     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      0     0     0
SBOX(b5) = 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      0     0     1

SBOX(b5) = 11010101 = d5

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com