Calculation of 0x58

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x58 =  01011000 = x^6 + x^4 + x^3

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

Calculation of 0x58^-1 in the finite field GF(2⁸)

00100011 = 00000001 * 100011011 + 00000101 * 1011000
00011110 = 00000010 * 100011011 + 00001011 * 1011000
00000001 = 00000111 * 100011011 + 00011000 * 1011000
00000000 = 01011000 * 100011011 + 100011011 * 1011000

a^-1(x) = x^4 + x^3 = 00011000 = 0x18

The calculation of 0x58^-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      0     1     1
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      1     0     1
SBOX(58) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      0     1     1
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      0     0     0

SBOX(58) = 01101010 = 6a

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com