International Association
for Cryptologic Research

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.