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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