Islamic State, which has hundreds of French-speaking fighters, claimed responsibility for the Champs Elysees shooting soon afterwards, in a statement identifying the attacker as "Abu Yousif al-Belgiki (the Belgian)".
The man, named in reports as Karim Cheurfi, got out of a vehicle on the Champs-Élysées on Thursday night and started shooting.
Mr Molins said Cheurfi had a long criminal record, including trying to attack police previously.
France has lived under a state of emergency since 2015 and has suffered a spate of Islamist militant attacks that have killed more than 230 people in the past two years.
US President Donald Trump tweeted that the attack "will have a big effect" on the election.More news: Freeport-McMoran Copper & Gold, Inc. (NYSE:FCX) Under Analyst Spotlight
Investigators searched a home early Friday in an eastern suburb of Paris believed linked to the attack and police detained for questioning three of the gunman's family members - routine in such cases.
A Belgium prosecutor said Friday the "man came to police late yesterday after he saw himself appear on social media as terror suspect Number One".
The attack unfolded on the famed Champs-Elysees around 9 p.m. local time (3 p.m. ET) when a auto stopped in front of a police van, according to French Interior Ministry spokesman Pierre Henry Brandet.
France's government has reviewed its already extensive election security measures and says it is "fully mobilized" in the wake of the Champs-Elysees gun attack on police officers.
Marty Cisse, an office cleaner from Mauritania, worries that Le Pen's closed-borders platform would threaten immigrants like himself, but said "security is important" and said it should be the priority of the next president.
ISIS quickly claimed responsibility for the attack.More news: Trump celebrates after his unpopularity puts red seats in play
The man responsible for a shootout in central Paris on Thursday night was a French national, Belgian interior minister Jan Jambon has told public broadcaster VRT.
French president Francois Hollande said he was convinced the circumstances of the attack pointed to a terrorist act.
The Thursday evening attack took place in the city's Champs-Elysees district, which is hugely popular with tourists and shoppers. Sometimes we suspect someone but can't prove anything so we're forced to release them.
The attacker emerged from a auto and used an automatic weapon to shoot at officers outside a Marks & Spencer's store at the center of the Champs-Elysees, Molins said.
Barack Obama spoke to Macron on Thursday about the "important upcoming presidential election in France", a spokesman for the former USA president said.
After killing an officer, the man attempted to run away while shooting at other officers, two of whom he injured, the spokesman added.More news: No one will be fired in United passenger dragging incident
The impact on the outcome of the French election is unclear - Sunday is the poll's first round - but far-right leader Marine Le Pen, her centrist rival Emmanuel Macron, and scandal-hit conservative Francois Fillon cancelled campaign events planned for today.
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