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Calculation of 0xe6

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xe6 =  11100110 = x^7 + x^6 + x^5 + x^2 + x
m(x) = (x + 1) * a(x) + (x^5 + x^4 + 1)

Calculation of 0xe6-1 in the finite field GF(28)

00110001 = 00000001 * 100011011 + 00000011 * 11100110
00010011 = 00000101 * 100011011 + 00001110 * 11100110
00000100 = 00001110 * 100011011 + 00010001 * 11100110
00000011 = 00111101 * 100011011 + 01001010 * 11100110
00000001 = 01001001 * 100011011 + 11001111 * 11100110
00000000 = 11100110 * 100011011 + 100011011 * 11100110

a-1(x) = x^7 + x^6 + x^3 + x^2 + x + 1 = 11001111 = 0xcf

The calculation of 0xe6-1 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     1
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     1
SBOX(e6) = 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      1     1     0
           0 0 0 1 1 1 1 1      1     0     1


SBOX(e6) = 10001110 = 8e

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com