A leader of the militant group Boko Haram has released a video claiming it carried out a series of attacks in north-east Nigeria over the Christmas period.

Abubakar Shekau, the head of one of Boko Haram’s three factions, declared the group active and operational a day after the Nigerian president said the militants had been defeated.

“We are in good health and nothing has happened to us,” he said in the video released on Tuesday. “Nigerian troops, police and those creating mischief against us can’t do anything against us, and you will gain nothing.

“We carried out the attacks in Maiduguri, in Gamboru, in Damboa. We carried out all these attacks.”

There has been a surge in violence in recent months, with dozens of people killed in suicide bombings and attacks on military bases in the region. Authorities have put this down to desperate insurgents trying to find food, weapons and ammunition.

Three people were reportedly burned to death on Christmas Day in Molai, a village near Maiduguri, the birthplace of Boko Haram. On Saturday 25 loggers were killed 13 miles from the state capital.

It is unclear to which Gamboru attack Shekau was referring, but he may have meant an ambush of an aid convoy on the Gamboru road in which four people were killed.

His declaration that the self-styled Islamists are alive and well contradicted the Nigerian president’s new year message which was broadcast the day before.

“We have since beaten Boko Haram,” said Muhammadu Buhari. “Isolated attacks still occur, but even the best-policed countries cannot prevent determined criminals from committing terrible acts of terror.”

It is unclear why he made such a claim, after his exhortation in 2015 that the Nigerian military had “technically won the war” against Boko Haram was met with derision.

In mid-December the government announced it was releasing $1bn from its excess oil account to pay for military equipment and training to fight the insurgency. Later it said not all the money would be spent on fighting Boko Haram alone.

Nigeria has recently announced a plan to combat Boko Haram by making villagers living in insecure areas move to garrison towns – although this was in effect how the military operated for most of 2017.

It is unclear how villagers who have always made their living farming, herding and fishing will be expected to survive when they are living on lands not their own and are not permitted to leave.

It is impossible to build up a clear picture of what exactly is happening in north-east Nigeria and across its borders in Cameroon and Chad.

According to a security tracker set up by the Council on Foreign Relations to monitor deaths reported in the media, more than 30,000 people have died in the conflict since May 2011. There are thought to have been large numbers of unreported deaths in inaccessible regions.

For the past two years, the Nigerian government has insisted it is winning or has won the war. According to the Nigerian army’s latest pronouncement, 700 Boko Haram captives recently escaped to Monguno, one of the local government areas of Borno state, after military operations in the region.

Pouring scorn on declarations of his death or capture, Shekau periodically releases video messages aimed at the government, the Nigerian people or a global audience. In the most notorious of these, he paraded the Chibok girls, a group of 300 schoolgirls kidnapped by his men, and vowed he would “sell them in the market”.