1,137 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
An introduction to the abelian Reidemeister torsion of three-dimensional manifolds
These notes accompany some lectures given at the autumn school "Tresses in
Pau" in October 2009. The abelian Reidemeister torsion for 3-manifolds, and its
refinements by Turaev, are introduced. Some applications, including relations
between the Reidemeister torsion and other classical invariants, are surveyed.Comment: 53 pages. Notes of a small course given at the autumn school "Tresses
in Pau" (October 2009). Minor modifications in this second versio
A New Cryptosystem Based On Hidden Order Groups
Let be a cyclic multiplicative group of order . It is known that the
Diffie-Hellman problem is random self-reducible in with respect to a
fixed generator if is known. That is, given and
having oracle access to a `Diffie-Hellman Problem' solver with fixed generator
, it is possible to compute in polynomial time (see
theorem 3.2). On the other hand, it is not known if such a reduction exists
when is unknown (see conjuncture 3.1). We exploit this ``gap'' to
construct a cryptosystem based on hidden order groups and present a practical
implementation of a novel cryptographic primitive called an \emph{Oracle Strong
Associative One-Way Function} (O-SAOWF). O-SAOWFs have applications in
multiparty protocols. We demonstrate this by presenting a key agreement
protocol for dynamic ad-hoc groups.Comment: removed examples for multiparty key agreement and join protocols,
since they are redundan
Global properties of tensor rank
AbstractThe dependence of tensor rank on the underlying ring of scalars is considered. It is shown that the integers are, in a certain sense, the worst scalars. A ring of scalars can be improved by adjoining algebraic elements but not by adjoining indeterminates. The real closed fields are the best scalars among ordered rings, and the algebraically closed fields are best among all rings. Let B(Rm×n×p) be the maximum tensor rank of any m×n×p array of elements from the ring R. A generalization of Gaussian elimination shows that B(Rn×n×n)⩽34n2 for most useful rings R. For every R, B(Rm×n×p)⩾mnp/(m+n+p), and slightly stronger lower bounds are proven for R a field
The homotopy type of the loops on -connected -manifolds
For we compute the homotopy groups of -connected closed
manifolds of dimension . Away from the finite set of primes dividing
the order of the torsion subgroup in homology, the -local homotopy groups of
are determined by the rank of the free Abelian part of the homology.
Moreover, we show that these -local homotopy groups can be expressed as a
direct sum of -local homotopy groups of spheres. The integral homotopy type
of the loop space is also computed and shown to depend only on the rank of the
free Abelian part and the torsion subgroup.Comment: Trends in Algebraic Topology and Related Topics, Trends Math.,
Birkhauser/Springer, 2018. arXiv admin note: text overlap with
arXiv:1510.0519
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