Why does only length prepending improve the security of CBC-MAC

Question

I know that length prepending improves security of CBC-MAC. However, wouldn't inserting the length elsewhere (middle, end or any other part of message) be equally good? After all, even the length is processed by the underlying cipher block.

Answer 1

Well, let's try it, and see how hard it is to forge a message.

Let's say for illustrative purposes that each character is a block, and that numbers represent the length indicator section. And let's start by putting the length indicator at the end. So,

XXXXXXX7

represents a 7-block message, with the '7' indicator at the end. Let's also say that,

$_{MAC}($XXXXXXX7$) = M_0$

represents the MACing of that 7-block message, with the resulting Tag $M_0$.

How hard is it to forge? It is trivially easy. All you need to do is request the MAC of three messages like so:

1) $_{MAC}($AAAAA5BBB9$) = M_1$

Note the '5' in the middle of the message. For this first MAC, that is just part of the requested message -- only the '9' at the end has been added by the length-appending algorithm to indicate the total length of the message.

2) $_{MAC}($AAAAA5$) = M_2$

3) $_{MAC}($CCCCC5$) = M_3$

Then xor $M_2$ and $M_3$:

$M_2 \oplus M_3 = D$

And let $ E = B \oplus D$, and now you can trivially forge the Tag for a new message that you did not request from the MAC Oracle:

$_{MAC}($CCCCC5EBB9$) = M_1$

Note that the same logic applies if the length indicator is anywhere in the message except right at the beginning.