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

• Published in: IEEE transactions on circuits and systems. II, Express briefs Express briefs, 2023-08, Vol.70 (8), p.3064-3068
• Main Authors: Souza, Carlos E. C., Moreno, Davi, Figueiredo, Ravi B. D., Chaves, Daniel P. B., Pimentel, Cecilio
• Format: Article
• Language: English
• Subjects: <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">q -analogs; Algebra; Analogs; Field programmable gate arrays; Fields (mathematics); Finite element analysis; finite fields; Galois fields; Generators; Hardware; Iterative methods; number theory; Permutations; Proposals; pseudo-random number generators; Pseudorandom; Random numbers
• ISSN: 1549-7747; 1558-3791
• DOI: 10.1109/TCSII.2023.3247242

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.