# Subscript R notation for the finite fields

I'm trying to understand the notation used in the literature for Pairing-based cryptography.

I know (and I hope I've understood it well) from Wikipedia that $\mathbb{Z}_p$ is the finite field of prime order $p$, where

• $p$ is the order of the field
• $q$ and the characteristic of the field and $q=p^n$

I came across this notation here and there while researching the Identity-Based Encryption (Boneh-Franklin): $$s \in_R\mathbb{Z}^*_q$$

The $\mathbb{Z}^*$ means that the finite field is provided with the multiplication operation.

But the $_R$ confuses me, as I can't find its meaning on the web.

Could somebody explain it?

PS: are the following notations equivalent? $GF(p)$, $\mathbb{Z}_p$, $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{F}_p$

• The $_R$ has nothing to do with the field — it is associated to $\in$! To quote your first link: "For a set $S$, by $a\in_RS$, we mean that $a$ is randomly chosen from $S$." – yyyyyyy Apr 17 '16 at 13:08
• Oh, shame on me ^^. And for the notation equivalence ? – EisenHeim Apr 17 '16 at 13:10
• If $p\in \mathbb P$ (with $\mathbb P$ being the set of all primes) then the notations $GF(p); \mathbb Z_p; \mathbb Z/p\mathbb Z; \mathbb F_p$ are equivalent. – SEJPM Apr 17 '16 at 13:16
• Note that depending on the context $\mathbb Z_p$ is also used for the $p$-adic integers. – flawr Apr 18 '16 at 9:54

The $_R$ has nothing to do with the field — it is associated to $\in$! To quote your first link: "For a set $S$, by $a\in_R S$, we mean that $a$ is randomly chosen from $S$."
If $p\in \mathbb P$ (with $\mathbb P$ being the set of all primes) then the notations $GF(p);\mathbb Z_p;\mathbb Z/p\mathbb Z;\mathbb F_p$ are equivalent.
• Another (realted) question : does $X =\langle U, V \rangle$ mean that $X$ is the concatenation of $U$ and $V$ ? – EisenHeim Apr 18 '16 at 7:08