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Calculation of 0x72

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x72 =  01110010 = x^6 + x^5 + x^4 + x
m(x) = (x^2 + x) * a(x) + (x^5 + x^4 + x^2 + x + 1)

Calculation of 0x72-1 in the finite field GF(28)

00110111 = 00000001 * 100011011 + 00000110 * 1110010
00011100 = 00000010 * 100011011 + 00001101 * 1110010
00001111 = 00000101 * 100011011 + 00011100 * 1110010
00000010 = 00001000 * 100011011 + 00110101 * 1110010
00000001 = 00111101 * 100011011 + 10010111 * 1110010
00000000 = 01110010 * 100011011 + 100011011 * 1110010

a-1(x) = x^7 + x^4 + x^2 + x + 1 = 10010111 = 0x97

The calculation of 0x72-1 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      1     0     0
           1 1 1 1 0 0 0 1      0     0     0
SBOX(72) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 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      1     0     0


SBOX(72) = 01000000 = 40

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com