Calculation of 0x85
m(x) = 0x11b = 100011011 = x^8 + x^4 + x^3 + x + 1
a(x) = 0x85 = 10000101 = x^7 + x^2 + 1
m(x) = (x) * a(x) + (x^4 + 1)
Calculation of 0x85-1 in the finite field GF(28)00010001 = 00000001 * 100011011 + 00000010 * 10000101
00001101 = 00001000 * 100011011 + 00010001 * 10000101
00000110 = 00011001 * 100011011 + 00110001 * 10000101
00000001 = 00111010 * 100011011 +
01110011 * 10000101
00000000 = 10000101 * 100011011 + 100011011 * 10000101
a
-1(x) = x^6 + x^5 + x^4 + x + 1 = 01110011 = 0x73
The calculation of 0x85
-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 1 1 1
1 1 1 0 0 0 1 1 0 0 1
1 1 1 1 0 0 0 1 0 0 0
SBOX(85) = 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 1 1 0
0 0 0 1 1 1 1 1 0 0 1
SBOX(85) = 10010111 = 97
For more information see
FIPS 197.
Implemented by bachph [at] philba [dot] com