Determination of a type of permutation trinomials over finite fields

Xiang-dong Hou

Xiang-dong Hou

Acta Arithmetica, Tome 166 (2014), p. 253-278 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $f = a x + b x^{q} + x^{2 q - 1} \in_{q} [x]$ . We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of $_{q ²}$ . This result allows us to solve a related problem: Let $g_{n, q} \in_{p} [x]$ (n ≥ 0, $p = c h a r_{q}$ ) be the polynomial defined by the functional equation $\sum_{c \in_{q}} {(x + c)}^{n} = g_{n, q} (x^{q} - x)$ . We determine all n of the form $n = q^{α} - q^{β} - 1$ , α > β ≥ 0, for which $g_{n, q}$ is a permutation polynomial of $_{q ²}$ .

Publié le : 2014-01-01

Zbl 06373520

EUDML-ID : urn:eudml:doc:279147

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     author = {Xiang-dong Hou},
     title = {Determination of a type of permutation trinomials over finite fields},
     journal = {Acta Arithmetica},
     volume = {166},
     year = {2014},
     pages = {253-278},
     zbl = {06373520},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa166-3-3}
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Xiang-dong Hou. Determination of a type of permutation trinomials over finite fields. Acta Arithmetica, Tome 166 (2014) pp. 253-278. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa166-3-3/