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Calculation of 0xb7

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0xb7 =  10110111 = x^7 + x^5 + x^4 + x^2 + x + 1
m(x) = (x) * a(x) + (x^6 + x^5 + x^4 + x^2 + 1)

Calculation of 0xb7-1 in the finite field GF(28)

01110101 = 00000001 * 100011011 + 00000010 * 10110111
00101000 = 00000011 * 100011011 + 00000111 * 10110111
00001101 = 00000100 * 100011011 + 00001011 * 10110111
00000110 = 00011011 * 100011011 + 00111101 * 10110111
00000001 = 00110010 * 100011011 + 01110001 * 10110111
00000000 = 10110111 * 100011011 + 100011011 * 10110111

a-1(x) = x^6 + x^5 + x^4 + 1 = 01110001 = 0x71

The calculation of 0xb7-1 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     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 0 0 1
SBOX(b7) = 1 1 1 1 1 0 0 0 * 1 + 0 = 0
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 1


SBOX(b7) = 10101001 = a9

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com