Post - Quantum Cryptography Using Complexity Doctoral Dissertation

In this thesis, we develop a cryptographic primitive: a solution to the secure set membership problem. The secure set membership problem is the problem of creating a set representation so that it is possible to verify that a string is a member of the set without being able to learn members of the set from the representation. The secure set membership primitive includes a distributed protocol for set establishment and a proof-of-possession protocol to show set membership without revealing the member of the set. We use a subset of the instances of the 3SAT witness-finding problem in developing a new complexity assumption for the secure set membership primitive. It is risky to base cryptosystems onNP-complete problems, as what is hard in the worst case may be easy in most cases. To mitigate this risk, we limit the instances of 3SAT that we generate to try to select only hard instances; our complexity assumption is different than mere NP-completeness. We assume that 3SAT instances with certain parameters are difficult to find witnesses for. We back up this assumption with experiments using the MiniSAT SAT solver as well as our own software. We use these experiments to determine which parameters are most likely to yield a secure system. We suggest applications of secure set membership including anonymous digital credentials, accounts with multiple users, access control lists, and document timestamping. We include both centralized and distributed protocols for establishing the set, and both pseudonomynous and anonymous protocols for verifying that a user possesses a witness that is a member of the set.