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Calculation of 0x08

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x8 =  00001000 = x^3
m(x) = (x^5 + x + 1) * a(x) + (x + 1)

Calculation of 0x08-1 in the finite field GF(28)

00000011 = 00000001 * 100011011 + 00100011 * 1000
00000001 = 00000111 * 100011011 + 11101000 * 1000
00000000 = 00001000 * 100011011 + 100011011 * 1000

a-1(x) = x^7 + x^6 + x^5 + x^3 = 11101000 = 0xe8

The calculation of 0x08-1 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 0 1 0
1 1 1 0 0 0 1 1 0 0 0
1 1 1 1 0 0 0 1 1 0 0
SBOX(08) = 1 1 1 1 1 0 0 0 * 0 + 0 = 1
0 1 1 1 1 1 0 0 1 1 1
0 0 1 1 1 1 1 0 1 1 0
0 0 0 1 1 1 1 1 1 0 0


SBOX(08) = 00110000 = 30

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com