Illegitimate child concealing his right identity

A child was born out of wedlock but he still uses his actual father's name everywhere to avoid embarrassment. He does that because only a few people know he is an illegitimate child. If he uses his mother's name this will lead to the suspicion of people and even those people who don't know that he is illegitimate will come to know about it. It wasn't the child's fault to go through this embarrassment. Is this fine and if not how should one go about to avoid people's taunts?