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This question is regarding "A Fully Secure Revocable ID-Based Encryption in the Standard Model" - Tung-Tso TSAI, Yuh-Min TSENG, Tsu-Yang WU

The encryption method specifies part of the encryption process to be $\hat e(g_1, g_2)^r \cdot M$ where $g_1$ and $g_2$ are public parameters in cyclic group $G_1$, $\hat e$ is the pairing function and $M$ is the actual message.

My question is what this $\cdot$ operator actually implies for an implementation - is it taking chunks of the $M$ equal to the bit length of the result of $\hat e(g_1, g_2)^r$ and then adding* them together?

*My understanding, perhaps wrongly, is that the multiply operation for elliptic curve groups is addition based on this

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Given that the authors (as any sane person should¹) use multiplicative notation for all of the involved groups, "$\cdot$" simply represents the group operation in the target group $G_2$.

It does not seem to be explicitly specified as far as I can tell from skimming the paper, but this implies that the message space of the encryption scheme is exactly $G_2$, meaning that we always have $M\in G_2$. So there are no "chunks" involved.

If you want to encrypt a message directly with the scheme, this means that you will need to encode it as a $G_2$ element. If your message is too long for that, you will need to use one of the standard domain extension techniques, such as using the RIBE as a KEM in a hybrid encryption scheme.


¹This is a hill I'm willing to die on.

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  • $\begingroup$ Thanks very helpful. $\endgroup$ – sumo Mar 24 at 0:35
  • $\begingroup$ Any thoughts on how to handle this with asymmetric pairing on a BN curve? With the implementation I'm using M will have to be in GT, is it best to get a random GT and feed the byte representation into a KDF? $\endgroup$ – sumo Apr 14 at 5:53
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    $\begingroup$ @sumo If you are using the encryption scheme as a KEM, then yes. Choose a uniform element $X\in G_T$ and encrypt it. Then use $KDF(X)$ as a key. $\endgroup$ – Maeher Apr 16 at 7:08

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