Question

I’m implementing AES in C# and at some point (MixColumns function) I have to multiply two Bytes over the GF(2^8) finite field.

So, I have three options:

• Use a default function that dotNet has (does it have something like that?)
• Write a custom function which does that
• Use lookup tables

For the custom function I found a piece of C code which I tried to rewrite for C#, but it doesn’t work (I get wrong results). (*)

Here is the original C piece of code (source):

/* Multiply two numbers in the GF(2^8) finite field defined 
 * by the polynomial x^8 + x^4 + x^3 + x + 1 */
uint8_t gmul(uint8_t a, uint8_t b) {
        uint8_t p = 0;
        uint8_t counter;
        uint8_t hi_bit_set;
        for (counter = 0; counter < 8; counter++) {
                if (b & 1) 
                        p ^= a;
                hi_bit_set = (a & 0x80);
                a <<= 1;
                if (hi_bit_set) 
                        a ^= 0x1b; /* x^8 + x^4 + x^3 + x + 1 */
                b >>= 1;
        }
        return p;
}

And this is what I rewrote:

public Byte GMul(Byte a, Byte b) { // Galois Field (256) Multiplication
   Byte p = 0;
   Byte counter;
   Byte hi_bit_set;
   for (counter = 0; counter < 8; counter++) {
      if ((b & 1) != 0) {
         p ^= a;
      }
      hi_bit_set = (Byte) (a & 0x80);
      a <<= 1;
      if (hi_bit_set != 0) {
         a ^= 0x1b; /* x^8 + x^4 + x^3 + x + 1 */
      }
      b >>= 1;
   }
   return p;
}

I also found some lookup tables here, and it seemed a simple and fine approach, but I don’t really know how to use them, though I got a hunch. (**)

Bottom line: which option should I choose, and how can I make it work, given what I wrote above is all I got so far, and that I don’t really want to go very deep with the math knowledge.

UPDATE:

*) Meanwhile I realised my C# rewrote code was producing correct answers, it was just my fault because I messed up when I verified them.

**) The tables can be used as a Byte[256] array, and the answer for, let’s say, x*3 is table_3[x], x being converted from HEX to DECIMAL when used as index for the table array.

Answer 1

In order to multiply x * 3 in GF(2), one just accesses x=table_3[x];

There’s probably a 3 Look-up-table method available that uses a logarithm approach.

Just as in regular numbers a*b = 2^(log2(a)+log2(b)), the same happens in GF(2), but without floating points or rounding errors.