The arrests were made in Montpellier, Clapiers and Marseillan after a two-week investigation by the anti-terrorist division of the Paris Public Prosecutor with the police and the national police force.
The 2016 law, approved after the attacks in Paris, was meant to stem the influence of jihadi social networks and online propaganda and made exceptions for purposes of research or informing the public.
Police raided the residence of a 20-year-old man in the southern French city of Montpellier and uncovered a makeshift laboratory filled with ingredients for making homemade explosives.
Last month, a soldier received minor injuries when a machete-wielding man tried to enter the Louvre museum in Paris. TATP was previously used in the November 2015 attack in Paris and the March 2016 attack in Brussels. The 29-year-old Egyptian was a resident of the United Arab Emirates and had a knife with him.
She recorded a video in which she pledged allegiance to so-called Islamic State (IS), M6 Info reports.
According to several reports out Friday, a 10-member constitutional council vetoed the law because it said it unnecessarily violated the people's freedom of communication.
Two of the four arrested were a 16-year-old girl and a young man that had already been under surveillance for ties to radical Islamists.
The girl, a convert to Islam, was arrested in Kundby, near the capital, in January previous year, and had been held in custody ever since.