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.