Information theoretic advances in zero-knowledge

Abstract

Zero-knowledge proofs have an intimate relation to notions from information theory. In particular, the class of all problems possessing statistical zero-knowledge proofs (SZK) was shown to have complete problems characterized by the statistical distance (Sahai and Vadhan [JACM, 20031) and entropy difference (Goldreich and Vadhan [CCC, 19991) of a pair of efficiently samplable distributions. This characterization has been extremely beneficial in understanding the computational complexity of languages with zero-knowledge proofs and deriving new applications from such languages. In this thesis, we further study the relation between zero-knowledge proofs and information theory. We show the following results: 1. Two additional complete problems for SZK characterized by other information theoretic notions-triangular discrimination and Jensen-Shannon divergence.

These new complete problems further expand the regime of parameters for which the STATISTICAL DIFFERENCE PROBLEM is complete for SZK. We further show that the parameterized STATISTICAL DIFFERENCE PROBLEM, for a regime of parameters in which this problem is not known to be in SZK, still share many properties with SZK. Specifically, its hardness implies the existence of one-way functions, and it and its complement have a constant-round public coin interactive protocol (i.e., AM n coAM). 2. The hardness of a problem related to the ENTROPY DIFFERENCE PROBLEM implies the existence of multi-collision resistant hash functions (MCRH). We also demonstrate the usefulness of such hash functions by showing that the existence of MCRH implies the existence of constant-round statistically hiding (and computationally binding) commitment schemes. 3. We initiate the study of zero-knowledge in the model of interactive proofs of proximity (IPP). We show efficient zero-knowledge IPPs for several problems.

We also show problems with efficient IPPs, for which every zero-knowledge IPP must be inefficient. Central in this study is showing that many of the statistical properties of SZK carry over to the IPP setting.