Perfect Zero Knowledge: New Upperbounds and Relativized Separations 📺
We investigate the complexity of problems that admit perfect zero-knowledge interactive protocols and establish new unconditional upper bounds and oracle separation results. We establish our results by investigating certain distribution testing problems: computational problems over high-dimensional distributions represented by succinct Boolean circuits. A relatively less-investigated complexity class SBP emerged as significant in this study. The main results we establish are: (1) A unconditional inclusion that $\NIPZK \subseteq \CoSBP$. (2) Construction of a relativized world in which there is a distribution testing problem that lies in NIPZK but not in SBP, thus giving a relativized separation of $\NIPZK$ (and hence PZK) from SBP. (3) Construction of a relativized world in which there is a distribution testing problem that lies in $\PZK$ but not in $\CoSBP$, thus giving a relativized separation of PZK$ from CoSBP. Results (1) and (3) imply an oracle separating PZK from NIPZK. Our results refine the landscape of perfect zero-knowledge classes in relation to traditional complexity classes.