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Calculation of 0x90

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x90 =  10010000 = x^7 + x^4
m(x) = (x) * a(x) + (x^5 + x^4 + x^3 + x + 1)

Calculation of 0x90-1 in the finite field GF(28)

00111011 = 00000001 * 100011011 + 00000010 * 10010000
00001010 = 00000110 * 100011011 + 00001101 * 10010000
00000111 = 00010101 * 100011011 + 00101100 * 10010000
00000011 = 00111001 * 100011011 + 01111001 * 10010000
00000001 = 01100111 * 100011011 + 11011110 * 10010000
00000000 = 10010000 * 100011011 + 100011011 * 10010000

a-1(x) = x^7 + x^6 + x^4 + x^3 + x^2 + x = 11011110 = 0xde

The calculation of 0x90-1 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     0
           1 1 0 0 0 1 1 1      1     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(90) = 1 1 1 1 1 0 0 0   *  1  +  0  =  0
           0 1 1 1 1 1 0 0      0     1     1
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      1     0     0


SBOX(90) = 01100000 = 60

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com