Subscript R notation for the finite fields
To quote yyyyyyy from the comments:


  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 $S$."


and to quote SEJPM from the comments:


  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.