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This whole question pertains to CBC-MAC.

Let's say $a$ is one cipher block long, and the attacker knows the MAC $M(a)$ of $a$, which was generated with a random key. How can the attacker forge the MAC of a two block message under the same key?

For example, if I have $M(a)$, can I forge $M(b||c)$, where $b$ and $c$ are both one block each?

If that is not possible, is it possible to forge $M(a||b)$ or $M(b||a)$?

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Given just $a$ and $M(a)$, you can construct the message $a\,\|\,b$, where $b = a \oplus M(a)$, which gives the same CBC-MAC output $M(a\,\|\,b) = M(a)$ as $a$ itself.

Indeed, we can append the block $b$ to the message as many times as we like without changing the MAC value: $M(a) = M(a\,\|\,b) = M(a\,\|\,b\,\|\,b) = M(a\,\|\,b\,\|\,b\,\|\,b) = \dots$


The reason this works should be apparent by looking at the structure of CBC-MAC:

CBC-MAC structure

As the diagram shows, a message like $a\,\|\,b\,\|\,b\,\|\,\dots$ is processed block by block. In the first step, we encrypt $a \oplus 0 = a$ with the block cipher, and obtain $M(a) = E_k(a)$ as the output. If the message ended here, this would be the resulting MAC value. However, if the message continues, the next block $b$ is XORed with the output $M(a)$ of the previous step and encrypted with the block cipher again. If we choose $b = a \oplus M(a)$, then the input to the block cipher will be $b \oplus M(a) = a \oplus M(a) \oplus M(a) = a \oplus 0 = a$ again, and thus the output will also be $E_k(a) = M(a)$ again.

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Welcome to Stackoverflow!

What you may be looking for https://en.wikipedia.org/wiki/CBC-MAC#Security_with_fixed_and_variable-length_messages

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