Hash proof systems

which was first introduced by Cramer and Shoup in 2002

is still one of the hottest research topics in cryptography.And then Wee proposed the concept of extractable hash proof system in 2010 and it is a concept extension on the hash proof system and as a paradigm of constructing PKE from search problems.On the basis of the extractable hash proof system

this paper expands the application scope of the extractable hash proof system by redefining the meaning of system parameters.We construct a basic All-But-One extractable hash proof system based on Diffie-Hellman relations by using the framework of extractable hash proof system.Based on this

fine-grained auxiliary input and weighting calculation are introduced.A new variable-policy CCA encryption scheme based on tag is proposed

and the security proof is also given in details.In particular

this scheme is a richer abstraction of extractable hash proof system that it is All-But-N

which means that the number of branches determined by the tag in the extraction mode could be n.At the same time

the scheme is based on the difficulty of the search problem and is essentially based on the computational Diffie-Hellman problem.