Calculation of 0x37

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x37 =  00110111 = x^5 + x^4 + x^2 + x + 1

m(x) = (x^3 + x^2 + x) * a(x) + (x^4 + 1)

Calculation of 0x37^-1 in the finite field GF(2⁸)

00010001 = 00000001 * 100011011 + 00001110 * 110111
00000100 = 00000011 * 100011011 + 00010011 * 110111
00000001 = 00001101 * 100011011 + 01000010 * 110111
00000000 = 00110111 * 100011011 + 100011011 * 110111

a^-1(x) = x^6 + x = 01000010 = 0x42

The calculation of 0x37^-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     1
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      0     0     1
SBOX(37) = 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     0
           0 0 0 1 1 1 1 1      0     0     1

SBOX(37) = 10011010 = 9a

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com