We require our systems to be complete, so honest executions will result in correct conclusions by the resolvers, sound, so malicious secondaries cannot cheat resolvers, and zero-knowledge, so resolvers will not learn additional information about elements they did not query explicitly. Providing proofs of membership is easy, as the primary can simply precompute signatures over all the members of the set. Providing proofs of non-membership, i.e. a denial-of-existence mechanism, is trickier and is the main issue in constructing PSR systems.
We provide three different strategies to construct a denial of existence mechanism. The first uses a set of cryptographic keys for all elements of the universe which are not members, which we implement using hierarchical identity based encryption and a tree based signature scheme. The second construction uses cuckoo hashing with a stash, where in order to prove non-membership, a secondary must prove that a search for it will fail, i.e. that it is not in the tables or the stash of the cuckoo hashing scheme. The third uses a verifiable ``random looking'' function which the primary evaluates over the set of members, then signs the values lexicographically and secondaries then use those signatures to prove to resolvers that the value of the non-member was not signed by the primary. We implement this function using a weaker variant of verifiable random/unpredictable functions and pseudorandom functions with interactive zero knowledge proofs.
For all three constructions we suggest fairly efficient implementations, of order comparable to other public-key operations such as signatures and encryption. The first approach offers perfect ZK and does not reveal the size of the set in question, the second can be implemented based on very solid cryptographic assumptions and uses the unique structure of cuckoo hashing, while the last technique has the potential to be highly efficient, if one could construct an efficient and secure VRF/VUF or if one is willing to live in the random oracle model.Category / Keywords: Zero-Knowledge, Efficiency, DNSSEC Date: received 2 Nov 2014, last revised 31 Mar 2015 Contact author: asafziv1987 at gmail com Available format(s): PDF | BibTeX Citation Version: 20150331:185235 (All versions of this report) Short URL: ia.cr/2014/905 Discussion forum: Show discussion | Start new discussion