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Calculation of 0xc8

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xc8 =  11001000 = x^7 + x^6 + x^3
m(x) = (x + 1) * a(x) + (x^6 + x + 1)

Calculation of 0xc8-1 in the finite field GF(28)

01000011 = 00000001 * 100011011 + 00000011 * 11001000
00001101 = 00000011 * 100011011 + 00000100 * 11001000
00000101 = 00010011 * 100011011 + 00111011 * 11001000
00000010 = 00110110 * 100011011 + 01001001 * 11001000
00000001 = 01111111 * 100011011 + 10101001 * 11001000
00000000 = 11001000 * 100011011 + 100011011 * 11001000

a-1(x) = x^7 + x^5 + x^3 + 1 = 10101001 = 0xa9

The calculation of 0xc8-1 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     0
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      1     0     1
SBOX(c8) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           0 1 1 1 1 1 0 0      1     1     1
           0 0 1 1 1 1 1 0      0     1     1
           0 0 0 1 1 1 1 1      1     0     1


SBOX(c8) = 11101000 = e8

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com