Officials said the criminal is seeing the victims' names in the news and then stealing their identities.
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Surfside Mayor Charles Burkett told WPLG that he is astonished and hopes that an arrest is soon made.
"(They) violated the rights of an individual, but worse than that, a deceased individual and a family that is grieving," he said.
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Families are urged to check on the credit of their loved ones to make sure they too were not targeted.
"It's terrible," Burkett said. "I can't wait to put a face to these deeds right now, and I think all of South Florida is eager to see who would do something like this - what kind of person would do something like this. But I'm looking forward to our police department apprehending them, and they are out there looking. I wouldn't want to be that person right now."
The mayor said he didn't want to disclose further details about how the hacker was able to steal the victims' identities because he did not believe it was helpful to release that information.
Authorities have not released further details about the crime or disclosed which victims were targeted.