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Determination of a type of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

Let f = a x + b x q + x 2 q - 1 ∈ 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 ∈ p [ x ] (n ≥ 0, p = c h a r q ) be the polynomial defined by the functional equation ∑ c ∈ 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 .

Facteurs communs et torsion en caractéristique non nulle

Laurent Denis (2011)

Journal de Théorie des Nombres de Bordeaux

Le pgcd de quantités de la forme a n - 1 et b n - 1 a été étudié dans différentes situations. Dans la première partie de ce texte nous prouverons que si a et b appartiennent à 𝔽 q [ T ] , le pgcd en question peut être borné indépendamment de n dans de nombreux cas. Ceci répond en particulier à une question de J. Silverman. Dans la deuxième partie nous étudierons un problème analogue dans la situation des modules de Drinfeld.

Left MQQs whose left parastrophe is also quadratic

Simona Samardjiska, Danilo Gligoroski (2012)

Commentationes Mathematicae Universitatis Carolinae

A left quasigroup ( Q , q ) of order 2 w that can be represented as a vector of Boolean functions of degree 2 is called a left multivariate quadratic quasigroup (LMQQ). For a given LMQQ there exists a left parastrophe operation q ∖ defined by: q ∖ ( u , v ) = w ⇔ q ( u , w ) = v that also defines a left multivariate quasigroup. However, in general, ( Q , q ∖ ) is not quadratic. Even more, representing it in a symbolic form may require exponential time and space. In this work we investigate the problem of finding a subclass of LMQQs whose left parastrophe...

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