Calculation of 0x75
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x75 = 01110101 = x^6 + x^5 + x^4 + x^2 + 1
m(x) = (x^2 + x) * a(x) + (x^5 + x^2 + 1)
Calculation of 0x75-1 in the finite field GF(28)00100101 = 00000001 * 100011011 + 00000110 * 1110101
00011010 = 00000011 * 100011011 + 00001011 * 1110101
00001011 = 00000100 * 100011011 + 00011011 * 1110101
00000111 = 00001111 * 100011011 + 00100110 * 1110101
00000010 = 00010101 * 100011011 + 01110001 * 1110101
00000001 = 00110000 * 100011011 +
10110101 * 1110101
00000000 = 01110101 * 100011011 + 100011011 * 1110101
a
-1(x) = x^7 + x^5 + x^4 + x^2 + 1 = 10110101 = 0xb5
The calculation of 0x75
-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 1 0 1
1 1 1 1 0 0 0 1 0 0 1
SBOX(75) = 1 1 1 1 1 0 0 0 * 1 + 0 = 1
0 1 1 1 1 1 0 0 1 1 0
0 0 1 1 1 1 1 0 0 1 0
0 0 0 1 1 1 1 1 1 0 1
SBOX(75) = 10011101 = 9d
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com