To describe the design approaches of IND-CCA2 (adaptive chosen ciphertext attack) secure public key encryption schemes systematically, the gaps between different kinds of intractable problems and IND-CCA2 security are studied. This paper points out that the construction of IND-CCA2 secure schemes is essentially to bridge these gaps. These gaps are categorized, analyzed and measured. Finally the methods to bridge these gaps are described. This explains the existing design approaches and gives an intuition about the difficulty of designing IND-CCA2 secure public key encryption schemes based on different types of assumptions.