Two topics in cryptography lattice problems and the security of protocols

Abstract: In this thesis we present new results in two areas – cryptographic protocols and lattice problems.• We present a new protocol for electronic cash which is designed to function on hardware with limited computing power. The scheme has provable security properties and low computational requirements, but it still gives a fair amount of privacy. Another feature of the system is that there is no master secret that could be used for counterfeiting money if stolen.• We introduce the notion of hierarchical group signatures. This is a proper generalization of group signatures, which allows multiple group managers organized in a tree with the signers as leaves. For a signer that is a leaf of the subtree of a group manager, the group manager learns which of its children that (perhaps indirectly) manages the signer. We provide definitions for the new notion and construct a scheme that is provably secure given the existence of a family of trapdoor permutations. We also present a construction which is relatively practical, and prove its security in the random oracle model under the strong RSA assumption and the DDH assumption.• We show a weakness in the specification for offline capable EMV payment cards. The weakness, which applies to cards without RSA capability, enables an attacker to duplicate a card and make transactions that cannot be tied to the original card.• We give a method for approximating any n-dimensional lattice with a lattice ? whose factor group Zn/? has n - 1 cycles of equal length with arbitrary precision. We also show that a direct consequence of this is that the Shortest Vector Problem and the Closest Vector Problem cannot be easier for this type of lattices than for general lattices.