The group said it was unlucky in its attack on a police station in Korydallos
An unknown Greek group has threatened to indiscriminately kill police officers and expand its target list to include prominent members of society.
The Sect of Revolutionaries issued the warning on a CD left on the grave of a teenager who was shot by police in December, triggering a month of riots.
Police said the declaration, which was published in a Greek newspaper on Thursday, appeared to be genuine.
The group also said it had attacked a police station in Athens on Tuesday.
Shots were fired at the station in the suburb of Korydallos, causing some damage but no injuries. A hand grenade was also thrown, but it failed to explode.
Last month, a policeman was shot and seriously wounded in central Athens. The attack was claimed by another better-known group, Revolutionary Struggle.
The proclamation by the Sect of Revolutionaries was published on Thursday morning in the newspaper, Ta Nea, just as a gas bomb exploded outside the political office of the public order ministry.
In it, the group said the armed attack on the police station in Korydallos had been in retaliation for the alleged murder by police of 15-year-old Alexis Grigoropoulos.
It claimed its members had been "unlucky" not to kill a police officer during the pre-dawn assault, which many had blamed on Revolutionary Struggle.
"Our aim was to execute them," the statement said. "Next time they will not have luck on their side."
From now on, the group warned, the life of a police officer would be worth as much as a bullet.
"The cops are like doughnuts - they're no good unless they have a hole in the middle," it said. "Start handing in your resignations, otherwise we will start measuring graves."
A police spokesman said it was taking the statement seriously.
"It seems to be genuine," Panayiotis Stathis told the Associated Press. "It's a group that has not appeared before but the methodology seems to be the same as that of Revolutionary Struggle."
The BBC's Malcolm Brabant in Athens says the desire for vengeance on behalf of the Sect of Revolutionaries and Revolutionary Struggle is worrying the police unions.
Greek police have protested against the repeated attacks of recent months
They have held demonstrations protesting against the violence, saying there are human beings behind the uniforms who are just as poorly paid as the average Greek.
Our correspondent says the militant groups appear to be modelling themselves on European urban guerrilla organisations from the 1970s, such as the German Red Army Faction.
The Sect of Revolutionaries also warned that prominent Greeks, including capitalists, state functionaries and journalists, would be targeted soon.
After three serious attacks on police in the past 40 days, officers are becoming stressed and understandably nervous, our correspondent adds.
On Wednesday, a policeman working near the US embassy shot and seriously injured a guard outside the ambassador's residence. The officer is reported to have claimed that he thought he was going to be shot.