Is it possible to convert a selective secure broadcast encryption scheme to an adaptive secure broadcast encryption scheme? I was reading the paper, entitled &quot;Multiparty Key Exchange, Efficient Traitor Tracing, and More from
Indistinguishability Obfuscation&quot;, by Boneh et al. On page 23, the authors have claimed the following.
&quot;Remark 6.2. We note that for private linear broadcast, there are only a polynomial number of
recipient sets, meaning selective and adaptive security are equivalent.&quot;
How can we prove this Remark? Is it possible to transform a selective secure private linear broadcast encryption scheme to an adaptive secure scheme (for the system which can accommodate at most a polynomial number of users)?
There is a generic reduction from breaking selective security of a protocol $\mathsf{P}$ to breaking its adaptive security: the adaptive reduction for $\mathsf{P}$ simply guesses (at random) the information $w$ (in your case, the recipient set) that the adversary commits to in the selective experiment. Therefore, if $w$ comes from a universe $\mathcal{W}$ (which in your case is the set of recipient sets) and if $\mathsf{P}$ is at least $\epsilon$-secure then $\mathsf{P}$ is also at least $\epsilon\cdot|\mathcal{W}|$-secure. Since the universe $\mathcal{W}$ in the case of private linear broadcast is only of polynomial size, the adaptive security of $\mathsf{P}$ follows.$^*$
I think this idea is folklore, but the earliest formal statement I found goes back to [BB] (Theorem 7.1); you can read more about it in [J+]
[BB] Boneh and Boyen, Efficient Selective-ID Secure Identity-Based Encryption Without Random Oracles, Eurocrypt 2004
[J+] Jafargholi et al.,  Be Adaptive, Avoid Overcommitting, Crypto 2017
$^*$ In most cases the universe is of exponential size and this lead to an exponential loss in security. But there are examples where this is the best we know!