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Calculation of 0xf8

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xf8 =  11111000 = x^7 + x^6 + x^5 + x^4 + x^3
m(x) = (x + 1) * a(x) + (x^4 + x + 1)

Calculation of 0xf8-1 in the finite field GF(28)

00010011 = 00000001 * 100011011 + 00000011 * 11111000
00001010 = 00001110 * 100011011 + 00010011 * 11111000
00000111 = 00011101 * 100011011 + 00100101 * 11111000
00000011 = 00101001 * 100011011 + 01111100 * 11111000
00000001 = 01001111 * 100011011 + 11011101 * 11111000
00000000 = 11111000 * 100011011 + 100011011 * 11111000

a-1(x) = x^7 + x^6 + x^4 + x^3 + x^2 + 1 = 11011101 = 0xdd

The calculation of 0xf8-1 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      0     1     0
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      1     0     0
SBOX(f8) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      0     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      1     0     0


SBOX(f8) = 01000001 = 41

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com