Calculation of 0xca

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xca =  11001010 = x^7 + x^6 + x^3 + x

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

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

01000101 = 00000001 * 100011011 + 00000011 * 11001010
00000101 = 00000011 * 100011011 + 00000100 * 11001010
00000001 = 00111101 * 100011011 + 01010011 * 11001010
00000000 = 11001010 * 100011011 + 100011011 * 11001010

a^-1(x) = x^6 + x^4 + x + 1 = 01010011 = 0x53

The calculation of 0xca^-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     0
           1 1 0 0 0 1 1 1      1     1     0
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      0     0     0
SBOX(ca) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      0     1     1
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      0     0     0

SBOX(ca) = 01110100 = 74

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com