The Multiple Equal-Difference Structure of Cyclotomic Cosets (2501.03516v1)

Abstract: In this paper we introduce the definition of equal-difference cyclotomic coset, and prove that in general any cyclotomic coset can be decomposed into a disjoint union of equal-difference subsets. Among the equal-difference decompositions of a cyclotomic coset, an important class consists of those in the form of cyclotomic decompositions, called the multiple equal-difference representations of the coset. There is an equivalent correspondence between the multiple equal-difference representations of $q$-cyclotomic cosets modulo $n$ and the irreducible factorizations of $X^{n}-1$ in binomial form over finite extension fields of $\mathbb{F}_{q}$. We give an explicit characterization of the multiple equal-difference representations of any $q$-cyclotomic coset modulo $n$, through which a criterion for $X^{n}-1$ factoring into irreducible binomials is obtained. In addition, we represent an algorithm to simplify the computation of the leaders of cyclotomic cosets.

• The paper introduces a method for decomposing cyclotomic cosets into multiple equal-difference subsets, linking their structure to binomial factorizations over finite fields.
• It establishes specific criteria for identifying cyclotomic cosets that possess this equal-difference structure based on divisibility properties.
• A practical algorithm is presented for determining leaders of cyclotomic cosets, leveraging their equal-difference property to enhance computational efficiency for applications like coding theory.

An Analytical Overview of "The Multiple Equal-Difference Structure of Cyclotomic Cosets"

The paper "The Multiple Equal-Difference Structure of Cyclotomic Cosets" by Zhu et al. significantly explores the decomposition of cyclotomic cosets into equal-difference subsets in a structured manner. Cyclotomic cosets are key to understanding the factorization of polynomials over finite fields, impacting areas such as coding theory and cryptography.

Main Contributions

The authors introduce the concept of equal-difference cyclotomic cosets and demonstrate that any cyclotomic coset can be decomposed into several equal-difference subsets, which is a non-trivial concept that links the structure of cyclotomic cosets to that of binomial factorizations over finite fields.

1. Equal-Difference Cyclotomic Cosets: A cyclotomic coset is termed to be of equal difference if it can be expressed as a complete arithmetic sequence within the bounds specified modulo $n$. The paper establishes criteria for identifying such cosets based on the divisibility properties concerning $q$ and $n$.
2. Multiple Equal-Difference Representations: Utilizing these equal-difference subsets, the paper details a method for constructing multiple equal-difference representations of cyclotomic cosets, taking advantage of $q^t$-cyclotomic decompositions. This approach reveals deeper insights into the structure of polynomial factorization.
3. Characterization and Criteria: An important theoretical development is the characterization of cyclotomic cosets in terms of their equal-difference representations. The authors derive specific conditions under which a $q$-cyclotomic coset can possess an equal-difference structure, paving the way for efficient computation of factors of $X^n - 1$.
4. Algorithm for Leaders: A practical contribution of the paper is an algorithm for determining leaders of cyclotomic cosets. This algorithm leverages the equal-difference property and enables more efficient computations in terms of representation and manipulation of cosets.

Implications and Speculation on Future Work

The implications of this research are multifold. Practically, by elucidating how cosets can be decomposed into equal-difference subsets, it fosters improved algorithms for computing constacyclic codes and related parameters efficiently. This could further affect the development of more secure and efficient cryptographic algorithms, given the relationship between cyclotomic cosets and residue class rings.

Theoretically, this work bridges the understanding of cyclotomic cosets with factorization into binomials over finite fields, which is valuable for simplifying complex polynomial expressions used in computational number theory. Further, the criteria for when a polynomial $X^n - 1$ can be factored into binomials over finite fields give insight into the structural properties of these polynomials.

Looking forward, the current line of inquiry opens pathways for exploration into algorithms that efficiently compute cyclotomic cosets and their equal-difference decompositions in higher-dimensional algebraic structures or over other finite fields with different properties. The robustness of the method in practical applications like coding theory, particularly for large-scale data, also merits further investigation.

In summary, the paper presents a rigorous, mathematically grounded exploration of cyclotomic cosets and their decompositions, providing tools and insights crucial for both theoretical exploration and practical application in the realms of coding theory, cryptography, and computational mathematics.