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

Abstract

In this work we define the q-analog, also known as q-deformation, of the finite field Zp, denoted as [Zp]q. We derive some algebraic properties of [Zp]q and establish the necessary conditions for which the q-analogs define a bijection between Zp and [Zp]q, or a permutation in Zp. 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.