Calculation of 0x97

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x97 =  10010111 = x^7 + x^4 + x^2 + x + 1

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

Calculation of 0x97^-1 in the finite field GF(2⁸)

00110101 = 00000001 * 100011011 + 00000010 * 10010111
00011100 = 00000111 * 100011011 + 00001111 * 10010111
00001101 = 00001111 * 100011011 + 00011100 * 10010111
00000110 = 00011001 * 100011011 + 00110111 * 10010111
00000001 = 00111101 * 100011011 + 01110010 * 10010111
00000000 = 10010111 * 100011011 + 100011011 * 10010111

a^-1(x) = x^6 + x^5 + x^4 + x = 01110010 = 0x72

The calculation of 0x97^-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     0
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      0     0     1
SBOX(97) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      1     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(97) = 10001000 = 88

For more information see FIPS 197.

Implemented by bachph [at] philba [dot] com