AES column mixing

[unregistered]

according to here
http://www.angelfire.com/biz7/atleast/mix_columns.pdf

in page 3, it is stated that

Now, we can add them together. As they are in binary form, addition will be using XOR.
r0 = {02.d4} + {03.bf} + {01.5d} + {01.30}
= 1011 0011 XOR 1101 1010 XOR 0101 1101 XOR 0011 0000
= 0000 0100
= 04 (in Hex)

When I add them manually, the result is 10 0001 1010, which is different from what I get when I xor them. It seems that the xor does not take any carries into account.

Since it is addition according to matrix multiplication, why is xor used instead?

[kbw]

When computers must add two numbers, the rule that: x xor y = (x + y) mod 2 for any two bits x and y allows for very fast calculation, as well.

taken from http://en.wikipedia.org/wiki/Binary_numeral_system

[unregistered]

but the xor does not take carries into account leading to a different result from that when I add the 2 numbers manually and taking the carry into account

[kbw]

That's correct, hence the y = (x + y) mod 2 and not y = (x + y).

[unregistered]

That's correct, hence the y = (x + y) mod 2 and not y = (x + y).

ahh so that means in the mix columns step of the official AES specification, the calculation should be implemented as y = (x + y) mod 2 and NOT y = (x + y)?

[kbw]

Dunno. But XOR's used here too: http://en.wikipedia.org/wiki/Rijndael_mix_columns