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Calculation of 0xa1

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0xa1 =  10100001 = x^7 + x^5 + 1
m(x) = (x) * a(x) + (x^6 + x^4 + x^3 + 1)

Calculation of 0xa1-1 in the finite field GF(28)

01011001 = 00000001 * 100011011 + 00000010 * 10100001
00010011 = 00000010 * 100011011 + 00000101 * 10100001
00000110 = 00001011 * 100011011 + 00010011 * 10100001
00000001 = 00110011 * 100011011 + 01111100 * 10100001
00000000 = 10100001 * 100011011 + 100011011 * 10100001

a-1(x) = x^6 + x^5 + x^4 + x^3 + x^2 = 01111100 = 0x7c

The calculation of 0xa1-1 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      0     1     1
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      1     0     0
SBOX(a1) = 1 1 1 1 1 0 0 0   *  1  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     1
           0 0 1 1 1 1 1 0      1     1     0
           0 0 0 1 1 1 1 1      0     0     0


SBOX(a1) = 00110010 = 32

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com