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I have a simple question about basic math. When I read different papers bout ABE, I see different methods to choose a random exponent.

What is the difference between the following options?

  1. Choose a random exponent $\,a \in \mathbb Z_p$.

  2. Choose a random exponent $\,a \in_{\small R} \mathbb Z_p$.

  3. Choose a random exponent $\,a \in \mathbb Z^*_p$.

  4. Choose a random exponent $\,a \in_{\small R} \mathbb Z^*_p$.

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    $\begingroup$ Where do you see these? Some authors may write $\in_R$ to mean ‘randomly chosen from’, while others use $\xleftarrow\$$ for that, but it's not a widely used standard notation, so it's hard to say without context. Most authors will define what they mean because it's not standard. (That said, $\mathbb Z_p^*$ pretty much always means the set of units in the ring $\mathbb Z/p\mathbb Z$ of integers modulo $p$, i.e. nonzero elements coprime with $p$, so it might be used without definition even though sometimes $\mathbb Z_p$ means the $p$-adic integers in other contexts.) $\endgroup$ – Squeamish Ossifrage Dec 1 '19 at 0:29
  • $\begingroup$ In reference ieeexplore.ieee.org/document/7448433?denied= I found that authors used β ∈ Zp, however in sciencedirect.com/science/article/pii/S0898122112001198, they used α ∈R Z∗p. $\endgroup$ – tesoke Dec 1 '19 at 1:22
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    $\begingroup$ In your second paper, on p. 1312, §2.1 ‘Notation’, it says: In this paper, $x \in_R S$ denotes the operation of picking an element $x$ at random and uniformly from a finite set $S$. $\endgroup$ – Squeamish Ossifrage Dec 1 '19 at 1:44
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    $\begingroup$ If $p$ is prime and large enough, the difference between picking $x$ uniformly at random from $\mathbb Z_p$ vs. $\mathbb Z_p^*$ is essentially immaterial because the probability of picking $0$ is negligible. $\endgroup$ – Squeamish Ossifrage Dec 1 '19 at 1:46
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    $\begingroup$ There is also $\underleftarrow{R}$ for picking uniformly. $\endgroup$ – kelalaka Dec 1 '19 at 8:15
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As the comments pointed out $a\in_{R} M$ means picking a sample $a$ uniformly from some set $M$. If a non-uniform distribution is implied, this will be stated.

In case you are also asking about it, $\mathbb{Z}_p^\ast$ is the multiplicative group (nonzero elements) of the integers modulo $p.$ In general $$ \mathbb{Z}_n^\ast=\{k: k \in \mathbb{Z}_n, \gcd(k,n)=1\} $$ is the multiplicative group over the integers modulo $n,$ made up of exactly the integers $k$ such that $\gcd(k,n)=1,$ and there are $\varphi(n)$ of them. Of course $\varphi(p)=p-1.$

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