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Anna-Maurin Graner

Irreducible polynomials over finite fields for coding and cryptography

Universität Rostock, 2024

https://doi.org/10.18453/rosdok_id00004733

Abstract: We present a closed explicit formula for all generating polynomials of a very popular set of codes - the constacyclic codes over finite fields. This problem is equivalent to the factorization of the polynomial X^n-a into monic irreducible factors over a finite field for all positive integers n. From our explicit formula for this factorization we also derive the factorization of the n-th cyclotomic polynomial and of any composition of the form f(X^n) where f is an irreducible polynomial. Additionally, we present a construction of a large set of irreducible polynomials of the same degree.

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