q-Analogs Over Finite Fields: Definition, Algebraic Properties, and Application in Pseudo-Random Number Generators

In this brief we define the
$q$
-analog, also known as
$q$
-deformation, of the finite field
$\mathbb {Z}_{p}$
, denoted as
$[\mathbb {Z}_{p}]_{q}$
. We derive some algebraic properties of
$[\mathbb {Z}_{p}]_{q}$
and establish the necessary conditions for which the
$q$
-analogs define a bijection between
$\mathbb {Z}_{p}$
and
$[\mathbb {Z}_{p}]_{q}$
, or a permutation in
$\mathbb {Z}_{p}$
. We employ the
$q$
-analogs over finite fields to design a pseudo-random number generator (PRNG) and its statistical properties are analyzed using the NIST test suite. The proposed PRNG is implemented in a field-programmable gate array (FPGA) and its hardware requirements are analyzed. The FPGA implementation shows that the pseudo-random number generator based on
$q$
-analogs exhibits a superior throughput while requiring less hardware resources in comparison with recent proposals in the literature.