Quantum-Accelerated Algorithms for Generating Random Primitive Polynomials Over Finite Fields

Primitive polynomials over finite fields are crucial resources with broad applications across various domains in computer science, including classical pseudo-random number generation, coding theory, and post-quantum cryptography. Nevertheless, the pursuit of an efficient classical algorithm for generating random primitive polynomials over finite fields remains an ongoing challenge. In this work, it shows how this problem can be solved efficiently with the help of quantum computers. Moreover, the designs of specific quantum circuits to implement them are also presented. The research paves the way for the rapid and real-time generation of random primitive polynomials in diverse quantum communication and computation applications. Primitive polynomials over finite fields are essential across various computer science domains, including pseudo-random number generation, coding theory, and post-quantum cryptography. Despite their significance, random generation of primitive polynomials remains a challenge for classical computers. This work presents two efficient quantum algorithms to address this task, which paves the way for future quantum communication and computation applications.image