2015 | OriginalPaper | Chapter
Statistical Randomized Encodings: A Complexity Theoretic View
Authors : Shweta Agrawal, Yuval Ishai, Dakshita Khurana, Anat Paskin-Cherniavsky
Published in: Automata, Languages, and Programming
Publisher: Springer Berlin Heidelberg
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A randomized encoding of a function
f
(
x
) is a randomized function
$$\hat{f}(x,r)$$
f
^
(
x
,
r
)
, such that the “encoding”
$$\hat{f}(x,r)$$
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
$$\mathsf {SRE} $$
SRE
of languages (or boolean functions) that admit an efficient statistical randomized encoding. That is,
$$\hat{f}(x,r)$$
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
$$\mathsf {SRE} $$
SRE
from efficient computation:
We give the first examples of promise problems and languages in
$$\mathsf {SRE} $$
SRE
that are widely conjectured to lie outside
$$\mathsf {P/poly}$$
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
$$\mathsf {SZK} $$
SZK
from
$$\mathsf {SRE} $$
SRE
:
We explore the relationship of
$$\mathsf {SRE} $$
SRE
with the class
$$\mathsf {SZK} $$
SZK
of problems possessing statistical zero knowledge proofs. It is known that
$$\mathsf {SRE} \subseteq \mathsf {SZK} $$
SRE
⊆
SZK
. We present an oracle separation which demonstrates that a containment of
$$\mathsf {SZK} $$
SZK
in
$$\mathsf {SRE} $$
SRE
cannot be proved via relativizing techniques.