# Do I need to prove this?

I am using ABE scheme that has already proven under BDHE assumption. Here is the scheme https://eprint.iacr.org/2008/290.pdf

In the key generation algorithm, I want to tie the user secret key components with a message $m∈GT$. $m$ is different from the one used in the encryption algorithm.
I did like this

$S=M.e(g^t, g^d) = M.e(g,g)^{td}$ where $t,d∈Z$ are randomly chosen. $t$ is used to tie all other secret key components together.

$S$ will be given to the user along with the secret key which have these components $(K=g^bg^{at}, L=g^t, h1^t, h2^t, …)$

$d$ and $g^d$ will be kept secure with the authority.

My question, do I need to re-prove the security of the scheme considering $S$

• What do you want that tieing to achieve? ​ ​ – user991 Jul 23 '16 at 20:48
• @RickyDemer thank you, I want to ensure that (1) it hard to distinguish the term $e(g,g)^{td}$? given the secret key including $S$, and (2) the user cannot exchange (collude) $S$ with other user's $S'$ – Alex Jul 23 '16 at 21:04
• "hard to distinguish the term $e(g,g)^{td}$" from what? ​ What does your property (2) mean? ​ ​ ​ ​ – user991 Jul 23 '16 at 23:01
• @RickyDemer (1) distinguish from a random value; the second property is about whether or not the scheme will be resisted against collusion between users – Alex Jul 23 '16 at 23:24
• How is $t$ generated for (1)? ​ ​ – user991 Jul 24 '16 at 0:25