51 research outputs found
Some results related to the conjecture by Belfiore and Sol\'e
In the first part of the paper, we consider the relation between kissing
number and the secrecy gain. We show that on an -dimensional even
unimodular lattice, if the shortest vector length is , then as the
number of vectors of length decreases, the secrecy gain increases. We will
also prove a similar result on general unimodular lattices. We will also
consider the situations with shorter vectors. Furthermore, assuming the
conjecture by Belfiore and Sol\'e, we will calculate the difference between
inverses of secrecy gains as the number of vectors varies. We will show by an
example that there exist two lattices in the same dimension with the same
shortest vector length and the same kissing number, but different secrecy
gains. Finally, we consider some cases of a question by Elkies by providing an
answer for a special class of lattices assuming the conjecture of Belfiore and
Sol\'e. We will also get a conditional improvement on some Gaulter's results
concerning the conjecture.Comment: This paper contains the note http://arxiv.org/abs/1209.3573. However,
there are several new results, including the results concerning a conjecture
by Elkie
A Classification of Unimodular Lattice Wiretap Codes in Small Dimensions
Lattice coding over a Gaussian wiretap channel, where an eavesdropper listens
to transmissions between a transmitter and a legitimate receiver, is
considered. A new lattice invariant called the secrecy gain is used as a code
design criterion for wiretap lattice codes since it was shown to characterize
the confusion that a chosen lattice can cause at the eavesdropper: the higher
the secrecy gain of the lattice, the more confusion. In this paper, a formula
for the secrecy gain of unimodular lattices is derived. Secrecy gains of
extremal odd unimodular lattices as well as unimodular lattices in dimension n,
16 \leq n \leq 23 are computed, covering the 4 extremal odd unimodular lattices
and all the 111 nonextremal unimodular lattices (both odd and even) providing
thus a classification of the best wiretap lattice codes coming from unimodular
lattices in dimension n, 8 < n \leq 23. Finally, to permit lattice encoding via
Construction A, the corresponding error correction codes are determined.Comment: 10 page
Formally Unimodular Packings for the Gaussian Wiretap Channel
This paper introduces the family of lattice-like packings, which generalizes
lattices, consisting of packings possessing periodicity and geometric
uniformity. The subfamily of formally unimodular (lattice-like) packings is
further investigated. It can be seen as a generalization of the unimodular and
isodual lattices, and the Construction A formally unimodular packings obtained
from formally self-dual codes are presented. Recently, lattice coding for the
Gaussian wiretap channel has been considered. A measure called secrecy function
was proposed to characterize the eavesdropper's probability of correctly
decoding. The aim is to determine the global maximum value of the secrecy
function, called (strong) secrecy gain.
We further apply lattice-like packings to coset coding for the Gaussian
wiretap channel and show that the family of formally unimodular packings shares
the same secrecy function behavior as unimodular and isodual lattices. We
propose a universal approach to determine the secrecy gain of a Construction A
formally unimodular packing obtained from a formally self-dual code. From the
weight distribution of a code, we provide a necessary condition for a formally
self-dual code such that its Construction A formally unimodular packing is
secrecy-optimal. Finally, we demonstrate that formally unimodular
packings/lattices can achieve higher secrecy gain than the best-known
unimodular lattices.Comment: Accepted for publication in IEEE Transactions on Information Theory.
arXiv admin note: text overlap with arXiv:2111.0143
Achieving Secrecy Capacity of the Gaussian Wiretap Channel with Polar Lattices
In this work, an explicit wiretap coding scheme based on polar lattices is
proposed to achieve the secrecy capacity of the additive white Gaussian noise
(AWGN) wiretap channel. Firstly, polar lattices are used to construct
secrecy-good lattices for the mod- Gaussian wiretap channel. Then we
propose an explicit shaping scheme to remove this mod- front end and
extend polar lattices to the genuine Gaussian wiretap channel. The shaping
technique is based on the lattice Gaussian distribution, which leads to a
binary asymmetric channel at each level for the multilevel lattice codes. By
employing the asymmetric polar coding technique, we construct an AWGN-good
lattice and a secrecy-good lattice with optimal shaping simultaneously. As a
result, the encoding complexity for the sender and the decoding complexity for
the legitimate receiver are both O(N logN log(logN)). The proposed scheme is
proven to be semantically secure.Comment: Submitted to IEEE Trans. Information Theory, revised. This is the
authors' own version of the pape
- …