The error message is not a syntax issue. It clearly says that the object you are trying to alter by adding your digital signature either doesn't exist or you don't have permission.
Error tells object does not exist because, as I said, I do not know how to reference database trigger in ADD SIGNATURE statement. Database trigger does not belong to any schema. To drop it we need to use ON DATABASE clause, otherwise we get same error:
DROP TRIGGER [database_trigger]
Msg 3701, Level 11, State 5, Line 1
Cannot drop the trigger 'database_trigger', because it does not exist or you do not have permission.
And there is no ON DATABASE clause in ADD SIGNATURE statement. If I try using it I get:
ADD SIGNATURE TO [database_trigger] ON DATABASE
BY CERTIFICATE [cert_database_trigger]
Msg 156, Level 15, State 1, Line 1
Incorrect syntax near the keyword 'ON'.
So again, question is how to reference database trigger in ADD SIGNATURE statement[/b].