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Suppose we have a tweakable block cipher ($E(K, T, X)$ and $D(K, T, X)$) that is used in a scheme where the tweak, either in whole or in part, is used as a counter to make each block encrypt to a distinct ciphertext. If we XOR all of the blocks together and then encrypt the result, do we have a secure tag?

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It is not secure if you use the $E(K, T, X)$ values as the exposed ciphertext; it is secure if you're using this scheme as a MAC (and use an independent scheme to encrypt).

If you send the $C_i = E(K, T_i, X_i)$ as the ciphertext (and send $E(K, 0, \bigoplus C_i)$ as the tag, then what the attacker could do is flip the same bit on two different $C_i$ values. The resulting $\bigoplus C_i$ will remain the same, and so the tag would validate (and the decrypted plaintext would be incorrect).

On the other hand, if you keep all the $C_i$ values internal, then this system can be viewed as a Carter-Wegman MAC. All you need to show is that $\bigoplus C_i$ is an almost universal hash function (which is fairly easy to do, assuming you get the message padding right), and the standard CW proofs apply.

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  • $\begingroup$ Ah. I see why this scheme isn't used. Each ciphertext block needs to be transformed again with a keyed function before XORing them all together and encrypting the result. Is there a limit to the length of a message that can be authenticated securely under such a scheme and how does it compare to a chained MAC like HMAC or CBC-MAC? $\endgroup$ – Melab Jan 8 '18 at 20:40
  • $\begingroup$ @Melab: no, there wouldn't be any particular length limitations (at least until you hit the birthday bound); the CW argument above shows that $\bigoplus C_i$ remains universal up until then (and it probably is beyond that as well, i don't have an immediate proof of that) $\endgroup$ – poncho Jan 8 '18 at 21:00
  • $\begingroup$ Right, the birthday bound. That's what I was talking about. I didn't know if it applied to XOR MACs. I've seen claims that schemes like Bernstein's "protected counter sum" have beyond birthday bound security. $\endgroup$ – Melab Jan 8 '18 at 21:02
  • $\begingroup$ Are XOR MACs a variant of Carter-Wegman MACs? $\endgroup$ – Melab Jan 8 '18 at 21:03
  • $\begingroup$ Yes (actually, not a 'variant', rather a 'type of CW MAC'); however your MAC is different than what is commonly termed an "XOR MAC" $\endgroup$ – poncho Jan 8 '18 at 21:12

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