@MISC{Ishai_statisticalrandomized, author = {Yuval Ishai and et al.}, title = {Statistical Randomized Encodings: A Complexity Theoretic View}, year = {} }

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Abstract

A randomized encoding of a function f(x) is a randomized function f̂(x, r), such that the “encoding ” f̂(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 SRE of languages (or boolean functions) that admit an efficient statistical randomized encoding. That is, f̂(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 SRE from efficient computation: We give the first examples of promise problems and languages in SRE that are widely conjectured to lie outside 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 SZK from SRE: We explore the relationship of SRE with the class SZK of problems possessing statistical zero knowledge proofs. It is known that SRE ⊆ SZK. We present an oracle separation which demonstrates that a containment of SZK in SRE cannot be proved via relativizing techniques.