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

Comparing balanced $\mathbb{Z}_v$-sequences obtained from ElGamal function to random balanced sequences