Permutation polynomials from a linearized decomposition

Abstract

In this paper we discuss the permutational property of polynomials of the form $f(L(x))+k(L(x))\cdot M(x)\in {\mathbb{F}}_{q^n}[x]$ over the finite field ${\mathbb{F}}_{q^n}$, where $L,M\in {\mathbb{F}}_q[x]$ are $q$-linearized polynomials. The restriction $L,M\in {\mathbb{F}}_q[x]$ implies a nice correspondence between the pair $(L,M)$ and the pair $(g,h)$ of conventional $q$-associates over ${\mathbb{F}}_q$ of degree at most $n-1$. In particular, by using the AGW criterion, permutational properties of our class of polynomials translates to some arithmetic properties of polynomials over ${\mathbb{F}}_q$, like coprimality. This relates the problem of constructing PPs of ${\mathbb{F}}_{q^n}$ to the problem of factorizing $x^n-1$ in ${\mathbb{F}}_q[x]$. We then specialize to the case where $L(x)$ is the trace polynomial from ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$, providing results on the construction of permutation and complete permutation polynomials, and their inverses. We further demonstrate that the latter can be extended to more general linearized polynomials of degree $q^{n-1}$.

11 pages, comments are welcome!