On linear representation, complexity and inversion of maps over finite fields

This paper defines a linear representation for nonlinear maps $F:F^n\to F^n$ where $F$ is a finite field, in terms of matrices over $F$. This linear representation of the map F associates a unique number N and a unique matrix M in $F^{N\times N}$, called the Linear Complexity and the Linear Representation of F respectively, and shows that the compositional powers $F^{\left(k\right)}$ are represented by matrix powers $M^k$. It is shown that for a permutation map F with representation M, the inverse map has the linear representation $M^{-1}$. This framework of representation is extended to a parameterized family of maps $F_{\lambda }\left(x\right):F\to F$, defined in terms of a parameter $\lambda \in F$, leading to the definition of an analogous linear complexity of the map $F_{\lambda }\left(x\right)$, and a parameter-dependent matrix representation $M_{\lambda }$ defined over the univariate polynomial ring $F\left[\lambda \right]$. Such a representation leads to the construction of a parametric inverse of such maps where the condition for invertibility is expressed through the unimodularity of this matrix representation $M_{\lambda }$. Apart from computing the compositional inverses of permutation polynomials, this linear representation is also used to compute the cycle structures of the permutation map. Lastly, this representation is extended to a representation of the cyclic group generated by a permutation map F, and to the group generated by a finite number of permutation maps over $F$.