Tests par lots rapides et privés et contributions aux mathématiques expérimentales

Abstract

This thesis is the culmination of research conducted between 2019 and 2023. It is divided into three parts. Inthe first part, we explore algorithms related to the Covid-19 pandemic, such as Pool Testing, a well-establishedtechnique where samples from multiple patients are pooled for collective testing, allowing for cost reduction and time savings. We propose algorithms taking into account the a priori probabilities that individual tests are positive, which can be evaluated during a prior clinical examination of the patient. We also examine Pool Testingin emergency situations, where certain samples need to be analyzed according to some prescribed priority order. In both cases, we propose new algorithms and analyze them in detail. This section also deals with DNA privacy preservation in Covid-19 tests. In the second part, we present our results in experimental mathematics, where we have discovered several new conjectures on continued fractions through automated exploration. All those conjectures have been numerically tested to assess their plausibility. Finally, the third part of this thesis is devoted to various results in the field of computer security, such as a previously unknown attack on the Mathematica software, a new protection mechanism against counterfeit medication, and new observations on zero-knowledge proofs.