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Calculation of 0x5e

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x5e =  01011110 = x^6 + x^4 + x^3 + x^2 + x
m(x) = (x^2 + 1) * a(x) + (x^5 + x^4 + x^3 + x^2 + 1)

Calculation of 0x5e-1 in the finite field GF(28)

00111101 = 00000001 * 100011011 + 00000101 * 1011110
00011001 = 00000011 * 100011011 + 00001110 * 1011110
00001111 = 00000111 * 100011011 + 00011001 * 1011110
00000111 = 00001101 * 100011011 + 00111100 * 1011110
00000001 = 00011101 * 100011011 + 01100001 * 1011110
00000000 = 01011110 * 100011011 + 100011011 * 1011110

a-1(x) = x^6 + x^5 + 1 = 01100001 = 0x61

The calculation of 0x5e-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     0
           1 1 1 0 0 0 1 1      0     0     0
           1 1 1 1 0 0 0 1      0     0     1
SBOX(5e) = 1 1 1 1 1 0 0 0   *  0  +  0  =  1
           0 1 1 1 1 1 0 0      1     1     0
           0 0 1 1 1 1 1 0      1     1     1
           0 0 0 1 1 1 1 1      0     0     0


SBOX(5e) = 01011000 = 58

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com