Discussion on the theoretical results of white-box cryptography

White-box cryptography (WBC) aims to resist attacks from attackers who can control all the implementation details of cryptographic schemes. In 2009, Saxena et al. proposed a fundamental of white-box cryptography via the notion "white-box property" (WBP). Under this model, they proved that there do not exist obfuscators that can satisfy every security notion for a program (the negative result). On the other hand, they proved that there exists an obfuscator satisfying WBP for some security notion (the positive result). These contributions provide us a general cognition of WBC, which is big progress for the theoretical research. To better understand them, we make discussion on each result and achieve some new results. For the negative result, we prove that insufficiently secure obfuscator is the real cause of the negative result. We point out that the security of a white-box scheme cannot be guaranteed if it is instantiated by a less secure obfuscator, since the obfuscator used in their proof does not satisfy the "Virtual Black-box Property" with auxiliary input. From our proof, we also conclude that the notion WBP is equal to "Virtual Black-box Property with auxiliary input". For the positive result, we prove that security notion under black-box model should not be used in white-box context without any modification; although the positive result is meaningful, it is unlikely to prove that an obfuscator satisfies WBP for IND-CPA, since the security notion "IND-CPA" is under black-box model, which has different adversary with WBP.