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Can someone help me understanding the CCM authenticity bound by Phillip Rogaway? i-e Success probability < $q_{\text{dec}}/2^\tau + \sigma/2^b$.

$\tau$ is tag length, where as $b$ is block length. $q_{\text{dec}}$ is the number of forgery attempts, I can not get whats "$\sigma$" is?

Also If i am not looking for forgery attempts but looking for the channel errors that cause invalid message to go through CBC MAC. Then for that case $q_{\text{dec}}=1$. Am I right? And I am attempting to send 500MB of data? So what will be the number of un-authenticated messages that go through CBC MAC because of channel errors.

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If you mean this paper by Rogaway, "Evaluation of Some Blockcipher Modes of Operation" (PDF, relevant page is 120) then it very clearly documents everything.

  • $q_{\text{dec}}$ is the total amount of decryption queries the adversary asks. If you only accept one decryption query before using different keys, this can be approximated by $1$.
  • $\sigma$ is the total number of block cipher calls the mode would make on the adversary's sequence of queries. So if all adversarial queries (encryption and decryption ones) together require 4 block cipher calls to process, then $\sigma=4$. For 500 MByte of Data (in the single query) this would imply $\sigma\approx 2^{26}$

Also note that the bound stated in the paper (note the additional square on the $\sigma$) is $$\leq \frac{q_{\text{dec}}}{2^\tau}+\frac{\sigma^2}{2^b}$$

Plugging in the above values yields an advantage of $\leq 1/2^\tau+2^{52}/2^b$ which bounds the success probability of any adversary. Now random noise usually isn't exactly a smart adversary so the probability of a (successful) forgery from random noise is even much lower.


So let's clarify on what a "block cipher call" actually is. Modes of operation like CCM are built on block-ciphers which get modelled as functions $C=E_K(P)$ for some plaintext $P$, some key $K$ and the corresponding ciphertext $C$ where $P$ and $C$ are the same, fixed size. One computation of $C$ from $K,P$ or of $P$ from $K,C$ is called one block cipher call (because in the implementation you call the block cipher function for this). Now for CCM mode, which I will skip describing here, you end up with about 2 block cipher calls per data block (one for privacy, one for authenticity), yielding the aforementioned $2^{26}\approx 2\cdot 500\cdot 10^6/16$.

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  • $\begingroup$ ThankYou SEJPM, It helps a lot. I am newbie to this world. I wasn't able to get the block cipher calls. Still I a have little bit confusion about single query and block cipher calls (if you can elaborate I will be thankful or refer me to any documentation). From your example it means 500MB = $(500*8*10^6)/128$ (AES block size)= $2^{25}$. So $\sigma$ is basically for a given amount of data how many blocks will be made (Depending upon block size). Thanks in advance $\endgroup$ Aug 3 '18 at 22:05
  • $\begingroup$ @faheemawan I have added a paragraph to (hopefully) clarify what a block cipher call is. $\endgroup$
    – SEJPM
    Aug 4 '18 at 8:56
  • $\begingroup$ ThankYou very much It was very helpful. I would just like to ask one last question. Hope you won't mind. If I am using Bluetooth Low energy(4.0) in which the transmitted frame size is 336 bytes. So for a single frame the bound will be $\leq \frac{1}{2^{32}} + (((336)/(16 * 2))^2)/2^{128}$, whereas if I consider an entire data of 500MB, then it will be like the previous case. Am i Right? Also if I only consider channel errors, so it will be quite less than that. Thanks in advance. $\endgroup$ Aug 4 '18 at 11:34
  • $\begingroup$ @faheemawan yes, if you make the reject / force-key-rotation decision on a per-frame basis, then the bound you gave is indeed the correct one (but you may want to note that $\frac{1}{2^{32}}\gg 2^{52}/2^{128}\gg 1764/2^{128}$ and as such the $\frac{1}{2^{32}}$ will be "your main worry"). $\endgroup$
    – SEJPM
    Aug 4 '18 at 11:42
  • $\begingroup$ Thanks, as TLS do it on per frame base. If I am not doing it on per frame base so $\frac{1}{2^{32}}$ will be increased further, then instead of one we have the number of attempts. Thanks SEJPM, discussion helps me absorb easily. $\endgroup$ Aug 4 '18 at 11:53

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