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Calculation of 0xe3

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xe3 =  11100011 = x^7 + x^6 + x^5 + x + 1
m(x) = (x + 1) * a(x) + (x^5 + x^4 + x^3 + x^2 + x)

Calculation of 0xe3-1 in the finite field GF(28)

00111110 = 00000001 * 100011011 + 00000011 * 11100011
00011011 = 00000100 * 100011011 + 00001101 * 11100011
00001000 = 00001001 * 100011011 + 00011001 * 11100011
00000011 = 00011111 * 100011011 + 00100110 * 11100011
00000001 = 01010100 * 100011011 + 11101011 * 11100011
00000000 = 11100011 * 100011011 + 100011011 * 11100011

a-1(x) = x^7 + x^6 + x^5 + x^3 + x + 1 = 11101011 = 0xeb

The calculation of 0xe3-1 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     0
           1 1 1 1 0 0 0 1      1     0     0
SBOX(e3) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      1     0     0


SBOX(e3) = 00010001 = 11

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com