A man claiming to be Abubakar Shekau made the comments in a video posted online on Saturday, saying the group attacked a bus station in retaliation for the what he described as the government's collusion with the United States in the killing of Muslims.
"This is a prelude," said the man, who wore camouflage and held an AK-47 assault rifle, in the video.
In a warning to Nigerian President Goodluck Jonathan, the man said: "Let me be blunt: I am in your city, near you. Find me."
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The video surfaced the same day that an education official said a Nigerian schoolgirl, who was among the 129 kidnapped Monday by suspected Boko Haram militants, had returned home.
Even so, 84 of the girl's classmates remain unaccounted for.
A total of 45 girls are now free, according to a statement from Borno State Education Commissioner Musa Inuwa Kulo.
This still unsettled situation began Monday night, when militants engaged in a battle with guards at the Government Girls Secondary School in the northeastern Nigerian town of Chibok began herding the girls onto buses and trucks and drove off, authorities said.
But some of the schoolgirls managed subsequently to escape, including 14 on Friday and others on previous days, according to Kulo.
This is a far cry from the Defense Ministry's previous report that all but eight of the girls had found freedom, a claim that Nigeria's military retracted Thursday.
This retraction spurred fervent criticism of the government, with Lawan Zanna -- the father of one of the students -- blasting it as having resorted to "blatant propaganda" by making a "blatant lie."
While the mass abduction is extraordinary, violence in the region has been on the rise.
That violence was on display Monday when a bomb exploded at the bus station in Abuja during rush hour, killing at least 75 people and wounding more than 100.
But nowhere has it been more prevalent that in northeastern Nigeria's Borno, Yobe and Adamawa provinces, which have been under a state of emergency since mid-May due to persistent bloodshed there the government blames on Boko Haram.
The Islamist militant group -- who name means "Western education is sin" in the local Hausa language -- has bombed schools, churches and mosques; kidnapped women and children; and assassinated politicians and religious leaders alike.
Boko Haram says its aim is to impose a stricter enforcement of Sharia law across Africa's most populous nation, which is split between a majority Muslim north and a mostly Christian south.
The advocacy group Human Rights Watch claims more than 3,000 people have been killed in Boko Haram-related violence in the past five years.
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Who is Shekau?
Under Shekau, who took control of the group in 2009, violence carried out by the group has flourished, according to authorities.
Questions have swirled about Shekau, including whether he's dead or alive. In recent years, the Nigerian military has touted his death, only to retract its claim after he appeared alive and vibrant in propaganda videos.
He uses the alias Darul Tawheed, and analysts describe him as a ruthless loner and master of disguise.
The United States has put a $7 million bounty on Shekau's head. It also designated Boko Haram as a foreign terrorist network last year.
The group is diffuse, and Skekau does not speak directly with members, opting to communicate through a few select confidants. This may be why Shekau made no mention of the kidnapping of the schoolgirls.