Properties of exponential transformations of~finite~field

A. A. Gruba

Prikladnaâ diskretnaâ matematika, no. 2 (2023), pp. 13-29

Résumé

We consider exponential transformations acting on the set $V_n(p)$ of all vectors of length $n$ over a prime field $P_0 = \text{GF}(p)$ ($p$ is a prime number). For every element $\gamma\in P = \text{GF}(p^n)$ with a minimal polynomial $F(x)$ of degree $n$ over the field $P_0$, consider the mapping $\hat{s} : P \rightarrow P$, where $\hat{s}(0) = 0$ and if $x \neq 0$, then $\hat{s}(x) = \gamma^{\sigma(x)}$, $\sigma : P \rightarrow \{0, 1,\ldots, p^n - 1\}$ is a mapping that matches each element $x\in P$ with the number $\sigma(x) = x_0 + px_1 + \ldots +p^nx_{n-1}$, $\mathbf{x} = (x_0, \ldots , x_{n-1})$ is given by its coordinates in the basis $\mathbf{\alpha}$ of the vector space $P_{P_0}$. Transformation $s = \tau^{-1}\cdot\hat{s}\cdot \varkappa$, where $\tau : P \rightarrow V_n(p)$ matches $x\in P$ to its set of coordinates in the basis $\mathbf{\alpha}$ of $P_{P_0}$ and the mapping $\varkappa : P \rightarrow V_n(p)$ matches $x$ to its set of coordinates in the dual basis $\mathbf{\beta}$ of the basis $\mathbf{\alpha}$, is called an exponential transformation. We prove estimates for the degree of nonlinearity for an exponential transformation $s$: $(p-1)\left(n - \lceil \log_p(n+1) \rceil\right) \leq \deg s \leq n(p-1) - 1$, where $\lceil z \rceil$ is the minimum integer greater or equal to $z$. It is proved that $\deg s = n(p - 1) - 1$ if and only if the system $\gamma/(\gamma - 1), (\gamma/(\gamma-1))^p, \ldots, (\gamma/(\gamma - 1))^{p^{n-1}}$ is a basis of the vector space $P_{P_0}$. We also study some properties of the linear and differential characteristics of the transformation $s$.

Détail

Keywords: finite fields, linear recurrence, difference characteristic, linear characteristic.