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2014•Journal of Discrete Mathematical Sciences and CryptographyRequires access

Companion Matrix and Recognition of Primitive Polynomial

Huawei Huang, Chang-Wen Peng

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Abstract

This paper studies the properties of companion matrices of primitive polynomials over finite field and discusses the recognition of a primitive polynomial by companion matrix. It is showed that if (pn - 1) / (p - 1) is a prime number then the recognition of a primitive poly-nomial over finite field Fp of degree n can reduce to the recognition of a primitive element of Fp in time O (n3 log p). When p > 2, this can reduce the complexity of the recognition of a primitive polynomial.

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What this paper is about

This paper studies the properties of companion matrices of primitive polynomials over finite field and discusses the recognition of a primitive polynomial by companion matrix. It is showed that if (pn - 1) / (p - 1) is a prime number then the recognition of a primitive poly-nomial over finite field Fp of degree n can reduce to the recognition of a primitive element of Fp in time O (n3 log p). When p > 2, this can reduce the complexity of the recognition of a primitive polynomial.

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Available abstract

This paper studies the properties of companion matrices of primitive polynomials over finite field and discusses the recognition of a primitive polynomial by companion matrix. It is showed that if (pn - 1) / (p - 1) is a prime number then the recognition of a primitive poly-nomial over finite field Fp of degree n can reduce to the recognition of a primitive element of Fp in time O (n3 log p). When p > 2, this can reduce the complexity of the recognition of a primitive polynomial.

Key concepts: Primitive element, Primitive polynomial, Finite field, Prime (order theory), Mathematics, Polynomial, Matrix (chemical analysis), Combinatorics

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