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Calculation of 0xd2

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xd2 =  11010010 = x^7 + x^6 + x^4 + x
m(x) = (x + 1) * a(x) + (x^6 + x^5 + x^3 + x^2 + 1)

Calculation of 0xd2-1 in the finite field GF(28)

01101101 = 00000001 * 100011011 + 00000011 * 11010010
00001000 = 00000010 * 100011011 + 00000111 * 11010010
00000101 = 00011011 * 100011011 + 00100000 * 11010010
00000010 = 00110100 * 100011011 + 01000111 * 11010010
00000001 = 01110011 * 100011011 + 10101110 * 11010010
00000000 = 11010010 * 100011011 + 100011011 * 11010010

a-1(x) = x^7 + x^5 + x^3 + x^2 + x = 10101110 = 0xae

The calculation of 0xd2-1 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      1     1     0
           1 1 1 0 0 0 1 1      1     0     1
           1 1 1 1 0 0 0 1      1     0     0
SBOX(d2) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     1
           0 0 1 1 1 1 1 0      0     1     0
           0 0 0 1 1 1 1 1      1     0     1


SBOX(d2) = 10110101 = b5

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com