Arithmetic constraints of polynomial maps through discrete logarithms

TL;DR

This paper studies the distribution and independence of discrete logarithm functions of polynomial maps over finite fields, revealing conditions under which these functions behave independently asymptotically.

Contribution

It introduces a natural multiplicative condition ensuring asymptotic independence of discrete log functions of polynomial maps over finite fields.

Findings

Functions are asymptotically independent under certain conditions

Provides applications linking to previous research

Analyzes distribution of discrete logarithm functions

Abstract

Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$ elements and let $\theta$ be a generator of the cyclic group $\mathbb F_q^*$. For each $a\in \mathbb F_q^*$, let $\log_{\theta} a$ be the unique integer $i\in \{1, \ldots, q-1\}$ such that $a=\theta^i$. Given polynomials $P_1, \ldots, P_k\in \mathbb F_q[x]$ and divisors $1<d_1, \ldots, d_k$ of $q-1$, we discuss the distribution of the functions $$F_{i}:y\mapsto \log_{\theta}P_i(y)\pmod {d_i}, $$ over the set $\mathbb F_q\setminus \cup_{i=1}^k\{y\in \mathbb F_q\,|\, P_i(y)=0\}$. Our main result entails that, under a natural multiplicative condition on the pairs $(d_i, P_i)$, the functions $F_i$ are asymptotically independent. We also provide some applications that, in particular, relates to past work.