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Calculation of 0xdc

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xdc =  11011100 = x^7 + x^6 + x^4 + x^3 + x^2
m(x) = (x + 1) * a(x) + (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)

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

01111111 = 00000001 * 100011011 + 00000011 * 11011100
00100010 = 00000010 * 100011011 + 00000111 * 11011100
00011001 = 00000111 * 100011011 + 00001010 * 11011100
00001001 = 00001011 * 100011011 + 00011001 * 11011100
00000010 = 00011010 * 100011011 + 00100001 * 11011100
00000001 = 01100011 * 100011011 + 10011101 * 11011100
00000000 = 11011100 * 100011011 + 100011011 * 11011100

a-1(x) = x^7 + x^4 + x^3 + x^2 + 1 = 10011101 = 0x9d

The calculation of 0xdc-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     1
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     0
SBOX(dc) = 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      0     1     0
           0 0 0 1 1 1 1 1      1     0     1


SBOX(dc) = 10000110 = 86

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com