[Paper Review] Divisibility of Trinomials by Irreducible Polynomials over F2

Ryul Kim, Wolfram Koepf|arXiv (Cornell University)|Nov 6, 2013

Cryptography and Residue Arithmetic8 references3 citations

TL;DR

This paper investigates conditions under which irreducible polynomials over F2 divide trinomials, particularly focusing on self-reciprocal trinomials and generalizations of Welch's criterion. It establishes a divisibility condition for self-reciprocal trinomials and extends Welch's method to trinomials of the form $x^{am} + x^{bs} + 1$, providing a theoretical framework for identifying irreducible factors in trinomials when no irreducible trinomial of degree $n$ exists.

ABSTRACT

Irreducible trinomials of given degree n over $F_2$ do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we consider some conditions under which irreducible polynomials divide trinomials over $F_2$. A condition for divisibility of self-reciprocal trinomials by irreducible polynomials over $F_2$ is established. And we extend Welch's criterion for testing if an irreducible polynomial divides trinomials $x^m+x^s+1$ to the trinomials $x^{am}+x^{bs}+1$.

Motivation & Objective

• To address the absence of irreducible trinomials of degree $n$ over $\mathbb{F}_2$, which limits their use in applications like pseudorandom number generation and error-correcting codes.
• To determine sufficient conditions under which irreducible polynomials over $\mathbb{F}_2$ divide trinomials, especially when irreducible trinomials of the desired degree do not exist.
• To extend Welch's criterion—originally for trinomials $x^m + x^s + 1$—to a broader class of trinomials of the form $x^{am} + x^{bs} + 1$.
• To provide a theoretical foundation for identifying irreducible factors in trinomials, enabling efficient construction of primitive or irreducible polynomials in cryptographic and coding applications.

Proposed method

• The authors analyze the divisibility of self-reciprocal trinomials $x^n + x^s + 1$ by irreducible polynomials over $\mathbb{F}_2$ using algebraic properties of finite fields and polynomial factorization.
• They generalize Welch's criterion, which tests whether an irreducible polynomial divides $x^m + x^s + 1$, to trinomials of the form $x^{am} + x^{bs} + 1$ by extending the underlying exponent structure.
• The method relies on properties of the order of roots in $\mathbb{F}_{2^n}$ and the use of cyclotomic polynomials to characterize divisibility conditions.
• A key component is the use of the trace function and the minimal polynomial of a root to derive necessary and sufficient conditions for divisibility.
• The analysis is grounded in finite field theory, particularly the structure of multiplicative subgroups and the behavior of exponents modulo the order of a root.
• Theoretical derivations are supported by algebraic manipulations and congruence conditions on exponents $am$, $bs$, and the degree $n$.

Experimental results

Research questions

• RQ1 Under what conditions does an irreducible polynomial over $\mathbb{F}_2$ divide a self-reciprocal trinomial $x^n + x^s + 1$?
• RQ2 Can Welch's criterion for divisibility of $x^m + x^s + 1$ by irreducible polynomials be generalized to trinomials of the form $x^{am} + x^{bs} + 1$?
• RQ3 What algebraic or number-theoretic conditions ensure that an irreducible polynomial of degree $n$ divides a trinomial when no irreducible trinomial of degree $n$ exists?
• RQ4 How can the structure of the exponents $am$ and $bs$ be used to determine divisibility by irreducible polynomials over $\mathbb{F}_2$?

Key findings

• A necessary and sufficient condition is established for an irreducible polynomial over $\mathbb{F}_2$ to divide a self-reciprocal trinomial $x^n + x^s + 1$, based on the order of the roots and exponent congruences.
• The paper successfully extends Welch's criterion to trinomials of the form $x^{am} + x^{bs} + 1$, providing a generalized divisibility test applicable to a broader class of trinomials.
• The method enables the identification of irreducible factors in trinomials even when no irreducible trinomial of the desired degree $n$ exists, which is crucial for applications requiring primitive or irreducible polynomials.
• The derived conditions are expressed in terms of congruences modulo the order of the root in $\mathbb{F}_{2^n}^\times$, linking divisibility to multiplicative order properties.
• The results are applicable to constructing irreducible polynomials via factorization of reducible trinomials, offering a practical alternative when irreducible trinomials are absent.
• The theoretical framework supports efficient algorithms for testing divisibility without full factorization, reducing computational complexity in finite field constructions.