Lightning protection system and an aircraft having such a system

A lightning protection system for a composite structure has an outer surface designed to be subject to an air flow. This system has an electrically conductive strip and fasteners from the strip to the structure.Since the structure has holes designed for the fasteners, at least some of these fasteners have an attachment element from the lightning to this structure. This attachment element is inserted into the hole with its top part approximately even with the outer surface of this structure, and the attachment element is in electrical contact with the electrically conductive strip. At least part of the electrically conductive strip is placed within the thickness of the structure and to be connected to the ground.