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The Complexity of Estimating MinEntropy
, 2012
"... Goldreich, Sahai, and Vadhan (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZKcomplete. We consider the analogous problem for estimating the minentropy and prove that it is SBPcomplete, even when restricted to 3l ..."
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Goldreich, Sahai, and Vadhan (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZKcomplete. We consider the analogous problem for estimating the minentropy and prove that it is SBPcomplete, even when restricted to 3local samplers. For logarithmicspace samplers, we observe that this problem is NPcomplete by a result of Lyngsø and Pedersen on hidden Markov models (JCSS 2002). 1
Locally Computable UOWHF with Linear Shrinkage ∗
"... We study the problem of constructing locally computable Universal OneWay Hash Functions (UOWHFs) H: {0, 1} n → {0, 1} m. A construction with constant output locality, where every bit of the output depends only on a constant number of bits of the input, was established by [Applebaum, Ishai, and Kush ..."
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We study the problem of constructing locally computable Universal OneWay Hash Functions (UOWHFs) H: {0, 1} n → {0, 1} m. A construction with constant output locality, where every bit of the output depends only on a constant number of bits of the input, was established by [Applebaum, Ishai, and Kushilevitz, SICOMP 2006]. However, this construction suffers from two limitations: (1) It can only achieve a sublinear shrinkage of n − m = n 1−ɛ; and (2) It has a superconstant input locality, i.e., some inputs influence a large superconstant number of outputs. This leaves open the question of realizing UOWHFs with constant output locality and linear shrinkage of n−m = ɛn, or UOWHFs with constant input locality and minimal shrinkage of n − m = 1. We settle both questions simultaneously by providing the first construction of UOWHFs with linear shrinkage, constant input locality, and constant output locality. Our construction is based on the onewayness of “random ” local functions – a variant of an assumption made by Goldreich (ECCC 2000). Using a transformation of [Ishai, Kushilevitz, Ostrovsky and Sahai, STOC 2008], our UOWHFs give rise to a digital signature scheme with a minimal additive complexity overhead: signing nbit messages with security parameter κ takes only O(n + κ) time instead of O(nκ) as in typical constructions. Previously, such signatures were only known to exist under an exponential hardness assumption. As an additional contribution, we obtain new locallycomputable hardness amplification procedures for UOWHFs that preserve linear shrinkage. 1
Statistical Randomized Encodings: A Complexity Theoretic View
"... 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 secur ..."
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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 complexitytheoretic 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 nonstandard variant of the Decisional Diffie Hellman (DDH) assumption and the “Abelian Subgroup Membership problem” (which generalizes QuadraticResiduosity 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. 1