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Calculation of 0xdd

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xdd =  11011101 = x^7 + x^6 + x^4 + x^3 + x^2 + 1
m(x) = (x + 1) * a(x) + (x^6 + x^5 + x^4 + x^3 + x^2)

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

01111100 = 00000001 * 100011011 + 00000011 * 11011101
00100101 = 00000010 * 100011011 + 00000111 * 11011101
00010011 = 00000111 * 100011011 + 00001010 * 11011101
00000011 = 00001100 * 100011011 + 00010011 * 11011101
00000001 = 01001111 * 100011011 + 11111000 * 11011101
00000000 = 11011101 * 100011011 + 100011011 * 11011101

a-1(x) = x^7 + x^6 + x^5 + x^4 + x^3 = 11111000 = 0xf8

The calculation of 0xdd-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      0     1     1
           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     0
SBOX(dd) = 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 0      1     1     1
           0 0 0 1 1 1 1 1      1     0     1


SBOX(dd) = 11000001 = c1

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com