Speaking at a news conference in Paris, Molins said Cheurfi aged 39 had been identified by documents found in the car used in the attack. The suspect used to live with his mother in the suburb of Chelles, east of Paris, since he was released from jail in 2015.
Molins said the shooter had arrived in a car, walked out of the passenger seat and opened fire with a Kalashnikov at around 8.47 p.m. local time (1847GMT) on a police patrol that had been dispatched to protect the Turkish cultural center at the landmark boulevard.
The assailant was later shot dead. Two police officers were injured in the assault, one of whom remains in hospital.
Molins said "a piece of paper found" next to the suspect's car had a "handwritten message defending the cause of Daesh." However, the chief prosecutor said Cheurfi was known as a criminal offender but had showed no signs of "radicalization".
"Karim Cheurfi was known to the police and the judiciary. In the course of 2017, he sought to obtain weapons and made statements suggesting that he wanted to kill policemen," Molins said, adding the shooter had a long criminal record that dates back to at least 2001; he served several years in prison for driving a stolen car and shooting a police officer and his brother during the pursuit.
During the slain suspect's earlier arrest, he had grabbed the gun of a policeman while in custody and tried to kill him. He received a 20-year jail term on three counts of attempted murder in the case, which was later reduced to 15 years on appeal in February 2005; he was then released early from prison in 2013.
Three months later, he committed a violent robbery which culminated in another car chase with police.
In July 2014, he was sentenced to four years in prison, but the last two years were suspended.
He was again released from jail in 2015.
Molins confirmed the arrest of three family members of Cheurfi for questioning and said the investigation is ongoing to identify any possible accomplices.
The attack came less than three days before French voters are due to participate in a presidential election.