26 research outputs found
A Lattice Coding Scheme for Secret Key Generation from Gaussian Markov Tree Sources
In this article, we study the problem of secret key generation in the
multiterminal source model, where the terminals have access to correlated
Gaussian sources. We assume that the sources form a Markov chain on a tree. We
give a nested lattice-based key generation scheme whose computational
complexity is polynomial in the number, N , of independent and identically
distributed samples observed by each source. We also compute the achievable
secret key rate and give a class of examples where our scheme is optimal in the
fine quantization limit. However, we also give examples that show that our
scheme is not always optimal in the limit of fine quantization.Comment: 10 pages, 3 figures. A 5-page version of this article has been
submitted to the 2016 IEEE International Symposium on Information Theory
(ISIT
Secure Compute-and-Forward in a Bidirectional Relay
We consider the basic bidirectional relaying problem, in which two users in a
wireless network wish to exchange messages through an intermediate relay node.
In the compute-and-forward strategy, the relay computes a function of the two
messages using the naturally-occurring sum of symbols simultaneously
transmitted by user nodes in a Gaussian multiple access (MAC) channel, and the
computed function value is forwarded to the user nodes in an ensuing broadcast
phase. In this paper, we study the problem under an additional security
constraint, which requires that each user's message be kept secure from the
relay. We consider two types of security constraints: perfect secrecy, in which
the MAC channel output seen by the relay is independent of each user's message;
and strong secrecy, which is a form of asymptotic independence. We propose a
coding scheme based on nested lattices, the main feature of which is that given
a pair of nested lattices that satisfy certain "goodness" properties, we can
explicitly specify probability distributions for randomization at the encoders
to achieve the desired security criteria. In particular, our coding scheme
guarantees perfect or strong secrecy even in the absence of channel noise. The
noise in the channel only affects reliability of computation at the relay, and
for Gaussian noise, we derive achievable rates for reliable and secure
computation. We also present an application of our methods to the multi-hop
line network in which a source needs to transmit messages to a destination
through a series of intermediate relays.Comment: v1 is a much expanded and updated version of arXiv:1204.6350; v2 is a
minor revision to fix some notational issues; v3 is a much expanded and
updated version of v2, and contains results on both perfect secrecy and
strong secrecy; v3 is a revised manuscript submitted to the IEEE Transactions
on Information Theory in April 201
Multiple Packing: Lower and Upper Bounds
We study the problem of high-dimensional multiple packing in Euclidean space.
Multiple packing is a natural generalization of sphere packing and is defined
as follows. Let and . A multiple packing is a
set of points in such that any point in lies in the intersection of at most balls of radius around points in . We study the multiple packing
problem for both bounded point sets whose points have norm at most
for some constant and unbounded point sets whose points are allowed to be
anywhere in . Given a well-known connection with coding theory,
multiple packings can be viewed as the Euclidean analog of list-decodable
codes, which are well-studied for finite fields. In this paper, we derive
various bounds on the largest possible density of a multiple packing in both
bounded and unbounded settings. A related notion called average-radius multiple
packing is also studied. Some of our lower bounds exactly pin down the
asymptotics of certain ensembles of average-radius list-decodable codes, e.g.,
(expurgated) Gaussian codes and (expurgated) spherical codes. In particular,
our lower bound obtained from spherical codes is the best known lower bound on
the optimal multiple packing density and is the first lower bound that
approaches the known large limit under the average-radius notion of
multiple packing. To derive these results, we apply tools from high-dimensional
geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new
results added and significant revision. This paper is one of the three parts.
The other two are arXiv:2211.04408 and arXiv:2211.0440
Multiple Packing: Lower Bounds via Infinite Constellations
We study the problem of high-dimensional multiple packing in Euclidean space.
Multiple packing is a natural generalization of sphere packing and is defined
as follows. Let and . A multiple packing is a
set of points in such that any point in lies in the intersection of at most balls of radius around points in . Given a well-known connection
with coding theory, multiple packings can be viewed as the Euclidean analog of
list-decodable codes, which are well-studied for finite fields. In this paper,
we derive the best known lower bounds on the optimal density of list-decodable
infinite constellations for constant under a stronger notion called
average-radius multiple packing. To this end, we apply tools from
high-dimensional geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new
results added and significant revision. This paper is one of the three parts.
The other two are arXiv:2211.04408 and arXiv:2211.0440
Multiple Packing: Lower Bounds via Error Exponents
We derive lower bounds on the maximal rates for multiple packings in
high-dimensional Euclidean spaces. Multiple packing is a natural generalization
of the sphere packing problem. For any and , a
multiple packing is a set of points in such that
any point in lies in the intersection of at most balls
of radius around points in . We study this problem
for both bounded point sets whose points have norm at most for some
constant and unbounded point sets whose points are allowed to be anywhere
in . Given a well-known connection with coding theory, multiple
packings can be viewed as the Euclidean analog of list-decodable codes, which
are well-studied for finite fields. We derive the best known lower bounds on
the optimal multiple packing density. This is accomplished by establishing a
curious inequality which relates the list-decoding error exponent for additive
white Gaussian noise channels, a quantity of average-case nature, to the
list-decoding radius, a quantity of worst-case nature. We also derive various
bounds on the list-decoding error exponent in both bounded and unbounded
settings which are of independent interest beyond multiple packing.Comment: The paper arXiv:2107.05161 has been split into three parts with new
results added and significant revision. This paper is one of the three parts.
The other two are arXiv:2211.04407 and arXiv:2211.0440
Some "Goodness" Properties of LDA Lattices
We study some structural properties of Construction-A lattices obtained from
low density parity check (LDPC) codes over prime fields. Such lattices are
called low density Construction-A (LDA) lattices, and permit low-complexity
belief propagation decoding for transmission over Gaussian channels. It has
been shown that LDA lattices achieve the capacity of the power constrained
additive white Gaussian noise (AWGN) channel with closest lattice-point
decoding, and simulations suggested that they perform well under belief
propagation decoding. We continue this line of work, and prove that these
lattices are good for packing and mean squared error (MSE) quantization, and
that their duals are good for packing. With this, we can conclude that codes
constructed using nested LDA lattices can achieve the capacity of the power
constrained AWGN channel, the capacity of the dirty paper channel, the rates
guaranteed by the compute-and-forward protocol, and the best known rates for
bidirectional relaying with perfect secrecy.Comment: 27 pages, 3 figures. Revised version submitted to the Problems of
Information Transmission in July 201
List Decoding Random Euclidean Codes and Infinite Constellations
We study the list decodability of different ensembles of codes over the real
alphabet under the assumption of an omniscient adversary. It is a well-known
result that when the source and the adversary have power constraints and
respectively, the list decoding capacity is equal to . Random spherical codes achieve constant list
sizes, and the goal of the present paper is to obtain a better understanding of
the smallest achievable list size as a function of the gap to capacity. We show
a reduction from arbitrary codes to spherical codes, and derive a lower bound
on the list size of typical random spherical codes. We also give an upper bound
on the list size achievable using nested Construction-A lattices and infinite
Construction-A lattices. We then define and study a class of infinite
constellations that generalize Construction-A lattices and prove upper and
lower bounds for the same. Other goodness properties such as packing goodness
and AWGN goodness of infinite constellations are proved along the way. Finally,
we consider random lattices sampled from the Haar distribution and show that if
a certain number-theoretic conjecture is true, then the list size grows as a
polynomial function of the gap-to-capacity
Modified Bethe Permanent of a Nonnegative Matrix
Currently the best deterministic polynomial-time algorithm for approximating the permanent of a non-negative matrix is based on minimizing the Bethe free energy function of a certain normal factor graph (NFG). In order to improve the approximation guarantee, we propose a modified NFG with fewer cycles, but still manageable function-node complexity; we call the approximation obtained by minimizing the function of the modified normal factor graph the modified Bethe permanent. For nonnegative matrices of size 3× 3, we give a tight characterization of the modified Bethe permanent. For non-negative matrices of size n× n with n≥ 3, we present a partial characterization, along with promising numerical results. The analysis of the modified NFG is also interesting because of its tight connection to an NFG that is used for approximating a permanent-like quantity in quantum information processing. © 2020 IEEE
Local Decode and Update for Big Data Compression
This paper investigates data compression that simultaneously allows local decoding and local update. The main result is a universal compression scheme for memoryless sources with the following features. The rate can be made arbitrarily close to the entropy of the underlying source, contiguous fragments of the source can be recovered or updated by probing or modifying a number of codeword bits that is on average linear in the size of the fragment, and the overall encoding and decoding complexity is quasilinear in the blocklength of the source. In particular, the local decoding or update of a single message symbol can be performed by probing or modifying on average a constant number of codeword bits. This latter part improves over previous best known results for which local decodability or update efficiency grows logarithmically with blocklength. © 1963-2012 IEEE