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Calculation of 0x62

m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) =   0x62 =  01100010 = x^6 + x^5 + x
m(x) = (x^2 + x + 1) * a(x) + (x^5 + x^4 + x^2 + 1)

Calculation of 0x62-1 in the finite field GF(28)

00110101 = 00000001 * 100011011 + 00000111 * 1100010
00001000 = 00000010 * 100011011 + 00001111 * 1100010
00000101 = 00001101 * 100011011 + 00100101 * 1100010
00000010 = 00011000 * 100011011 + 01000101 * 1100010
00000001 = 00111101 * 100011011 + 10101111 * 1100010
00000000 = 01100010 * 100011011 + 100011011 * 1100010

a-1(x) = x^7 + x^5 + x^3 + x^2 + x + 1 = 10101111 = 0xaf

The calculation of 0x62-1 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      1     1     1
           1 1 1 0 0 0 1 1      1     0     0
           1 1 1 1 0 0 0 1      1     0     1
SBOX(62) = 1 1 1 1 1 0 0 0   *  0  +  0  =  0
           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(62) = 10101010 = aa

For more information see FIPS 197.



Implemented by bachph [at] philba [dot] com