44 research outputs found
2-Server PIR with sub-polynomial communication
A 2-server Private Information Retrieval (PIR) scheme allows a user to
retrieve the th bit of an -bit database replicated among two servers
(which do not communicate) while not revealing any information about to
either server. In this work we construct a 1-round 2-server PIR with total
communication cost . This improves over the
currently known 2-server protocols which require communication and
matches the communication cost of known 3-server PIR schemes. Our improvement
comes from reducing the number of servers in existing protocols, based on
Matching Vector Codes, from 3 or 4 servers to 2. This is achieved by viewing
these protocols in an algebraic way (using polynomial interpolation) and
extending them using partial derivatives
Query-Efficient Locally Decodable Codes of Subexponential Length
We develop the algebraic theory behind the constructions of Yekhanin (2008)
and Efremenko (2009), in an attempt to understand the ``algebraic niceness''
phenomenon in . We show that every integer ,
where , and are prime, possesses the same good algebraic property as
that allows savings in query complexity. We identify 50 numbers of this
form by computer search, which together with 511, are then applied to gain
improvements on query complexity via Itoh and Suzuki's composition method. More
precisely, we construct a -query LDC for every positive
integer and a -query
LDC for every integer , both of length , improving the
queries used by Efremenko (2009) and queries used by Itoh and
Suzuki (2010).
We also obtain new efficient private information retrieval (PIR) schemes from
the new query-efficient LDCs.Comment: to appear in Computational Complexit
On the Inner Product Predicate and a Generalization of Matching Vector Families
Motivated by cryptographic applications such as predicate encryption, we consider the problem of representing an arbitrary predicate as the inner product predicate on two vectors. Concretely, fix a Boolean function P and some modulus q. We are interested in encoding x to x_vector and y to y_vector so that P(x,y) = 1 = 0 mod q, where the vectors should be as short as possible. This problem can also be viewed as a generalization of matching vector families, which corresponds to the equality predicate. Matching vector families have been used in the constructions of Ramsey graphs, private information retrieval (PIR) protocols, and more recently, secret sharing.
Our main result is a simple lower bound that allows us to show that known encodings for many predicates considered in the cryptographic literature such as greater than and threshold are essentially optimal for prime modulus q. Using this approach, we also prove lower bounds on encodings for composite q, and then show tight upper bounds for such predicates as greater than, index and disjointness
On Embeddings of l_1^k from Locally Decodable Codes
We show that any -query locally decodable code (LDC) gives a copy of
with small distortion in the Banach space of -linear forms on
, provided and where , , and the distortion are simple functions of the code
parameters. We exhibit the copy of by constructing a basis for it
directly from "smooth" LDC decoders. Based on this, we give alternative proofs
for known lower bounds on the length of 2-query LDCs. Using similar techniques,
we reprove known lower bounds for larger . We also discuss the relation with
an alternative proof, due to Pisier, of a result of Naor, Regev, and the author
on cotype properties of projective tensor products of spaces
Proof of the Kakeya set conjecture over rings of integers modulo square-free
A Kakeya set is a set containing a
line in each direction. We show that, when is any square-free integer, the
size of the smallest Kakeya set in is at least
for any -- resolving a special
case of a conjecture of Hickman and Wright. Previously, such bounds were only
known for the case of prime . We also show that the case of general can
be reduced to lower bounding the rank of the incidence matrix of
points and hyperplanes over
Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree
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Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes