Statistical Randomized Encodings: A Complexity Theoretic View

Abstract

A randomized encoding of a function f(x) is a randomized function $\hat{f}(x,r)$, such that the “encoding” $\hat{f}(x,r)$ reveals f(x) and essentially no additional information about x. Randomized encodings of functions have found many applications in different areas of cryptography, including secure multiparty computation, efficient parallel cryptography, and verifiable computation.

We initiate a complexity-theoretic study of the class $\mathsf {SRE}$ of languages (or boolean functions) that admit an efficient statistical randomized encoding. That is, $\hat{f}(x,r)$ can be computed in time poly(|x|), and its output distribution on input x can be sampled in time poly(|x|) given f(x), up to a small statistical distance.

We obtain the following main results.

• Separating $\mathsf {SRE}$ from efficient computation: We give the first examples of promise problems and languages in $\mathsf {SRE}$ that are widely conjectured to lie outside $\mathsf {P/poly}$. Our candidate promise problems and languages are based on the standard Learning with Errors (LWE) assumption, a non-standard variant of the Decisional Diffie Hellman (DDH) assumption and the “Abelian Subgroup Membership problem” (which generalizes Quadratic-Residuosity and a variant of DDH).

• Separating $\mathsf {SZK}$ from $\mathsf {SRE}$ : We explore the relationship of $\mathsf {SRE}$ with the class $\mathsf {SZK}$ of problems possessing statistical zero knowledge proofs. It is known that $\mathsf {SRE} \subseteq \mathsf {SZK}$. We present an oracle separation which demonstrates that a containment of $\mathsf {SZK}$ in $\mathsf {SRE}$ cannot be proved via relativizing techniques.

Y. Ishai–Research supported by the European Union’s Tenth Framework Programme (FP10/2010-2016) under grant agreement no. 259426 ERC-CaC, ISF grants 1361/10 and 1709/14 and BSF grant 2012378.