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Calculation of 0x39

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x39 =  00111001 = x^5 + x^4 + x^3 + 1
m(x) = (x^3 + x^2 + 1) * a(x) + (x^3 + x^2 + x)

Calculation of 0x39-1 in the finite field GF(28)

00001110 = 00000001 * 100011011 + 00001101 * 111001
00000001 = 00000100 * 100011011 + 00110101 * 111001
00000000 = 00111001 * 100011011 + 100011011 * 111001

a-1(x) = x^5 + x^4 + x^2 + 1 = 00110101 = 0x35

The calculation of 0x39-1 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 0 1 1
1 1 1 0 0 0 1 1 1 0 0
1 1 1 1 0 0 0 1 0 0 0
SBOX(39) = 1 1 1 1 1 0 0 0 * 1 + 0 = 1
0 1 1 1 1 1 0 0 1 1 0
0 0 1 1 1 1 1 0 0 1 0
0 0 0 1 1 1 1 1 0 0 0


SBOX(39) = 00010010 = 12

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com