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Factorization and irreducibility of composed products

Lukas Kölsch, Lucas Krompholz, Gohar M. Kyureghyan

Abstract

Brawley and Carlitz introduced diamond products of elements of finite fields and associated composed products of polynomials in 1987. Composed products yield a method to construct irreducible polynomials of large composite degrees from irreducible polynomials of lower degrees. We show that the composed product of two irreducible polynomials of degrees $m$ and $n$ is again irreducible if and only if $m$ and $n$ are coprime and the involved diamond product satisfies a special cancellation property, the so-called conjugate cancellation. This completes the characterization of irreducible composed products, considered in several previous papers. More generally, we give precise criteria when a diamond product satisfies conjugate cancellation. For diamond products defined via bivariate polynomials, we prove simple criteria that characterize when conjugate cancellation holds. We also provide efficient algorithms to check these criteria. We achieve stronger results as well as more efficient algorithms in the case that the polynomials are bilinear. Lastly, we consider possible constructions of normal elements using composed products and the methods we developed.

Factorization and irreducibility of composed products

Abstract

Brawley and Carlitz introduced diamond products of elements of finite fields and associated composed products of polynomials in 1987. Composed products yield a method to construct irreducible polynomials of large composite degrees from irreducible polynomials of lower degrees. We show that the composed product of two irreducible polynomials of degrees and is again irreducible if and only if and are coprime and the involved diamond product satisfies a special cancellation property, the so-called conjugate cancellation. This completes the characterization of irreducible composed products, considered in several previous papers. More generally, we give precise criteria when a diamond product satisfies conjugate cancellation. For diamond products defined via bivariate polynomials, we prove simple criteria that characterize when conjugate cancellation holds. We also provide efficient algorithms to check these criteria. We achieve stronger results as well as more efficient algorithms in the case that the polynomials are bilinear. Lastly, we consider possible constructions of normal elements using composed products and the methods we developed.
Paper Structure (12 sections, 26 theorems, 78 equations, 2 algorithms)

This paper contains 12 sections, 26 theorems, 78 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Factorization of composed products of polynomials
  3. Irreducible diamond products of polynomials
  4. Decomposition of f⬦g over Fq
  5. Characterization and construction of diamond products satisfying conjugate cancellation
  6. Two key lemmas
  7. An important class of phi-products
  8. Algorithmic verification of conjugate cancellation
  9. Sufficient criteria for conjugate cancellation
  10. Diamond products from linearized polynomials
  11. Equivalent and sufficient criteria for conjugate cancellation
  12. When is ɸ(α,β) a normal element?

Key Result

Theorem 2.1

Let $m,n$ be two positive integers. Then the system of congruences has solutions if and only if $\gcd(m,n)\mid a_1-a_2$. Under this condition, the above system has only one solution modulo $\mathop{\mathrm{lcm}}\nolimits(m,n)$.

Theorems & Definitions (55)

  • Definition 1.1: Conjugate Cancellation on $G$
  • Theorem 2.1: Generalized Chinese Remainder Theorem
  • Lemma 2.2
  • Definition 2.3: Conjugate Cancellation on arbitrary sets
  • Example 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • proof
  • Theorem 2.7: Generalized Brawley--Carlitz--Theorem
  • ...and 45 more