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**2014/659**

**Title:**

On the Primitivity of Trinomials over Small Finite Fields

**Authors:**
YUjuan Li, Jinhua Zhao, Huaifu Wang

**Abstract:**

In this paper, we explore the primitivity of trinomials over small finite fields. We extend the results of the primitivity of trinomials x^{n}+ax+b over {\mathbb{F}}_{4} \cite{Li} to the general form x^{n}+ax^{k}+b. We prove that for given n and k, one of all the trinomials x^{n}+ax^{k}+b with b being the primitive element of {\mathbb{F}}_{4} and a+b\neq1 is primitive over {\mathbb{F}}_{4} if and only if all the others are primitive over {\mathbb{F}}_{4}. And we can deduce that if we find one primitive trinomial over {\mathbb{F}}_{4}, in fact there are at least four primitive trinomials with the same degree. We give the necessary conditions if there exist primitive trinomials over {\mathbb{F}}_{4}. We study the trinomials with degrees n=4^{m}+1 and n=21\cdot4^{m}+29, where m is a positive integer. For these two cases, we prove that the trinomials x^{n}+ax+b with degrees n=4^{m}+1 and n=21\cdot4^{m}+29 are always reducible if m>1. If some results are obviously true over {\mathbb{F}}_{3}, we also give it.

**ePrint:**
https://eprint.iacr.org/2014/659

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