Pseudo-Deterministic Construction of Irreducible Polynomials over Finite Fields

TL;DR

This paper introduces a polynomial-time pseudo-deterministic algorithm for constructing irreducible polynomials over finite fields, improving upon previous deterministic methods by leveraging randomized factoring algorithms.

Contribution

It extends Shoup's deterministic algorithm to a pseudo-deterministic version using fast randomized polynomial factoring, achieving better efficiency.

Findings

Runs in time O(d^4   q)

Uses randomized factoring to achieve pseudo-determinism

Improves efficiency over deterministic algorithms

Abstract

We present a polynomial-time pseudo-deterministic algorithm for constructing irreducible polynomial of degree $d$ over finite field $\mathbb{F}_q$. A pseudo-deterministic algorithm is allowed to use randomness, but with high probability it must output a canonical irreducible polynomial. Our construction runs in time $\tilde{O}(d^4 \log^4{q})$. Our construction extends Shoup's deterministic algorithm (FOCS 1988) for the same problem, which runs in time $\tilde{O}(d^4 p^{\frac{1}{2}} \log^4{q})$ (where $p$ is the characteristic of the field $\mathbb{F}_q$). Shoup had shown a reduction from constructing irreducible polynomials to factoring polynomials over finite fields. We show that by using a fast randomized factoring algorithm, the above reduction yields an efficient pseudo-deterministic algorithm for constructing irreducible polynomials over finite fields.