Having been convicted 27 times for robbery and other crimes, Chekkat was sent to prison between 2013 and 2015.
Nunez said the suspect became radicalized during one of his many stints in prison and was known to security services.
A source at the prosecutor's office said the motive behind the shooting was not immediately clear and that an investigation was under way to see if it was terrorism-related.
Meanwhile, the Paris prosecutor's office said a fifth person had been arrested and placed in custody in connection with the investigation.
Witnesses described to the AP hearing gunshots, screams and the shouts of police officers ordering people to stay indoors before the area fell silent and the officers fanned out.
Anupong Suebsamarn, 45, was the owner of a noodle factory in Chachoengsao province, east of Bangkok, and also sold clothes in the Thai capital's garment district, according to the English-language website of the newspaper Khao Sod.
Previously, French authorities had said the assailant killed three people, but Mr Heitz said two people were confirmed dead while the third was brain dead.
Armed French police have begun an operation in Strasbourg's Neudorf neighbourhood, a Reuters photographer said, as they hunt for a gunman who killed three people in an attack on a Christmas market. This latest incident comes three years after major terror attacks in Paris, and as France is recovering from several weeks of violent protests over the high cost of living. Italian daily La Repubblica reports that he was in Strasbourg to follow the European Parliamentary session.
Junior German interior minister Laurent Nuñez said on Wednesday that it could not be excluded from possibility that Chekatt had already crossed into Germany and border controls between the countries have been increased, according to The Guardian.
Who Is Cherif C? Strasbourg Christmas Market Gunman Information
Tajani said that the EU would "not be intimidated" by the attack and that European Parliamentary procedures would be set to go on this week as per usual.
Four people connected to him were detained in Strasbourg overnight, he added.
The Christmas market, a hugely popular attraction in the historic city, remained closed on Thursday. "It's this celebration that was hit yesterday by a terrorist act", he said.
Some lit candles and brought flowers to a makeshift memorial at the site of the attack.
In a bid to placate the protesters, French President Emmanuel Macron pledged €10 billion ($11.3 billion) in concessions to the poor. "The best thing would be to find him as quickly as possible", government spokesman Benjamin Griveaux told CNews.
Some Strasbourg residents have reported on social media that they heard gunfire in some parts of the city center.
At Chekatt's apartment, in an outer neighbourhood of Strasbourg, the lock of the door was broken at his apartment and police were guarding the building.
Seven of the injured are said to be in a serious condition.
Mr Castaner said: "Three hundred and fifty police and gendarmes are now on the ground to apprehend the suspect, supported by two helicopters, the RAID (French anti-terror police), the BRI (anti-gang brigade) and the Sentinel force". They found explosive materials, he said. 'You can just tell, ' said one, lightly touching the side of his head.
Saudi Energy Minister Says he Discussed Oil Market with US Counterpart
The contributions from OPEC and non-OPEC will correspond to 800,000 barrels per day and 400,000 barrels per day, respectively. Oil prices have fallen sharply since October on signs of an economic slowdown, with Brent losing nearly 30 percent in value.