"A suicide bomber riding a motorcycle detonated his explosives in a market in Asadabad the provincial capital, 10 people were killed and over 40 were injured," Ghani Mosamem, spokesman for the provincial governor, told AFP.
Sayed Maqsood Pacha, deputy provincial police chief, put the death toll slightly higher at 11.
No group immediately claimed responsibility for the attack, but Taliban insurgents, who have been waging a 14 year-long bloody insurgency in the country, have been blamed for such attacks in the past.
A witness to the blast, who did not want to be named, told AFP that an influential tribal elder, Haji Khan Jan, was the target of the attack and was killed.
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Jan had led a local uprising against the Taliban in the past, the witness added.
Kunar is a restive and remote province that shares a long border with neighbouring Pakistan - which the Afghan government has accused in the past of harbouring Taliban militants.
Delegates from Afghanistan, China, the US and Pakistan met in the Kabul last week for a fourth round of talks aimed at forming a path back to the nascent peace process, which was interrupted by last summer's announcement that Taliban leader Mullah Omar had died.