  In this paper, we investigate the randomness properties of sequences in
$\mathbb{Z}_v$ derived from permutations in $\mathbb{Z}_{p}^*$ using the
remainder function modulo $v$, where $p$ is a prime integer. Motivated by
earlier studies with a cryptographic focus we compare sequences constructed
from the ElGamal function $x \to g^x$ for $x\in\mathbb{Z}_{>0}$ and $g$ a
primitive element of $\mathbb{Z}_{p}^*$, to sequences constructed from random
permutations of $\mathbb{Z}_{p}^*$. We prove that sequences obtained from
ElGamal have maximal period and behave similarly to random permutations with
respect to the balance and run properties of Golomb's postulates for
pseudo-random sequences. Additionally we show that they behave similarly to
random permutations for the tuple balance property. This requires some
significant work determining properties of random balanced periodic sequences.
In general, for these properties and excepting for very unlikely events, the
ElGamal sequences behave the same as random balanced sequences.
