Sur la complexit\'e de familles d'ensembles pseudo-al\'eatoires

TL;DR

This paper investigates the complexity of families of pseudo-random subsets over finite fields, providing bounds and results based on exponential sums, combinatorics, and additive number theory.

Contribution

It introduces a new complexity measure for pseudo-random subsets and derives bounds using various mathematical techniques depending on the sets considered.

Findings

Bounds on the largest integer k for subset families

Results vary with the shape of sets S and P

Methods include exponential sums and additive combinatorics

Abstract

In this paper we are interested in the following problem. Let $p$ be a prime number, $S\subset \F_p$ and $\cP\subset \{P\in\F_p [X]:\deg P\le d\}$. What is the largest integer $k$ such that for all subsets $\cA, \cB$ of $\F_p$ satisfying $\cA\cap\cB =\emptyset$ and $|\cA\cup\cB |=k$, there exists $P\in\cP$ such that $P(x)\in S$ if $x\in\cA$ and $P(x)\not\in S$ if $x\in\cB$? This problem corresponds to the study of the complexity of some families of pseudo-random subsets. First we recall this complexity definition and the context of pseudo-random subsets. Then we state the different results we have obtained according to the shape of the sets $S$ and $\cP$ considered. Some proofs are based on upper bounds for exponential sums or characters sums in finite fields, other proofs use combinatorics and additive number theory.