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Pseudo-Deterministic Construction of Irreducible Polynomials over Finite Fields

Shanthanu S Rai

TL;DR

It is shown that by using a fast randomized factoring algorithm, the above reduction yields an efficient pseudo-deterministic algorithm for constructing irreducible polynomials over finite fields.

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.

Pseudo-Deterministic Construction of Irreducible Polynomials over Finite Fields

TL;DR

It is shown that by using a fast randomized factoring algorithm, the above reduction yields an efficient pseudo-deterministic algorithm for constructing irreducible polynomials over finite fields.

Abstract

We present a polynomial-time pseudo-deterministic algorithm for constructing irreducible polynomial of degree over finite field . 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 . Our construction extends Shoup's deterministic algorithm (FOCS 1988) for the same problem, which runs in time (where is the characteristic of the field ). 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.
Paper Structure (13 sections, 13 theorems, 7 equations, 3 algorithms)

This paper contains 13 sections, 13 theorems, 7 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Overview
  3. Preliminaries
  4. Pseudo-deterministic algorithms
  5. Finite Field primer
  6. Splitting field
  7. Structure of Finite Fields
  8. Conjugates and Minimal polynomial
  9. Representing finite field elements
  10. Equal degree polynomial factorization
  11. Construction of irreducible polynomials over extension fields $\mathbb{F}_{p^k}$
  12. Construction of irreducible polynomials over $\mathbb{F}_p$
  13. Conclusion

Key Result

theorem 1.2

There is a randomized polynomial-time algorithm $B$ such that, for infinitely many values of $n$, $B(1^n)$ outputs a canonical $n$-bit prime $p_n$ with high probability.

Theorems & Definitions (24)

  • definition 1.1
  • theorem 1.2
  • theorem 1.3
  • example 2.1
  • theorem 3.1: Existence and Uniqueness of Finite Fields
  • theorem 3.2: Fermat's little theorm for finite fields
  • theorem 3.3: Subfield Criterion
  • lemma 3.4
  • definition 3.5
  • lemma 3.6: Minimal polynomial of $\beta \in \mathbb{F}_{q^n}$
  • ...and 14 more