After correcting a misunderstanding regarding the security of additive PH and defining efficient PH, the authors show that any algebraic (or additive or multiplicative) PH can be at most IND-CCA1 secure in the black box model, but it still remains an open question whether IND-CCA1 security is a tight upper bound for algorithm-based PH. They believe it is extremely hard to find such efficient PHs as they have put some restrictions: homomorphic functions are length preserving. The additively homomorphic function can-not be addition or any other linear function.