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Calculation of 0x80

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x80 =  10000000 = x^7
m(x) = (x) * a(x) + (x^4 + x^3 + x + 1)

Calculation of 0x80-1 in the finite field GF(28)

00011011 = 00000001 * 100011011 + 00000010 * 10000000
00000010 = 00001110 * 100011011 + 00011101 * 10000000
00000001 = 01000111 * 100011011 + 10000011 * 10000000
00000000 = 10000000 * 100011011 + 100011011 * 10000000

a-1(x) = x^7 + x + 1 = 10000011 = 0x83

The calculation of 0x80-1 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     1
           1 1 0 0 0 1 1 1      1     1     0
           1 1 1 0 0 0 1 1      0     0     1
           1 1 1 1 0 0 0 1      0     0     1
SBOX(80) = 1 1 1 1 1 0 0 0   *  0  +  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     1


SBOX(80) = 11001101 = cd

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com