326 research outputs found
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
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
NFLlib: NTT-based Fast Lattice Library
International audienceRecent years have witnessed an increased interest in lattice cryptography. Besides its strong security guarantees, its simplicity and versatility make this powerful theoretical tool a promising competitive alternative to classical cryptographic schemes. In this paper, we introduce NFLlib, an efficient and open-source C++ library dedicated to ideal lattice cryptography in the widely-spread polynomial ring Zp[x]/(x n + 1) for n a power of 2. The library combines al-gorithmic optimizations (Chinese Remainder Theorem, optimized Number Theoretic Transform) together with programming optimization techniques (SSE and AVX2 specializations, C++ expression templates, etc.), and will be fully available under the GPL license. The library compares very favorably to other libraries used in ideal lattice cryptography implementations (namely the generic number theory libraries NTL and flint implementing polynomial arithmetic, and the optimized library for lattice homomorphic encryption HElib): restricting the library to the aforementioned polynomial ring allows to gain several orders of magnitude in efficiency
Tail-biting Codes for Lattice Wiretap Coding
The secrecy gain of Construction A lattices obtained by tail-biting rate 1/2 convolutional codes is studied to evaluate the secrecy performance of a lattice in a wiretap channel communication. The higher the secrecy gain, the more difficult for an eavesdropper to correctly guess the transmitted message. To explore the best potential secrecy gain, an exhaustive search is conducted, considering various constraint lengths and even code lengths up to 100. The methods' validity is verified by evaluating the minimum free distance and comparing the results with known findings. This investigation identifies outcomes close to the upper bound of unimodular lattices.Masteroppgave i informatikkINF399MAMN-PROGMAMN-IN
Dimension reduction techniques for the minimization of theta functions on lattices
We consider the minimization of theta functions
amongst
lattices , by reducing the dimension of the
problem, following as a motivation the case , where minimizers are
supposed to be either the BCC or the FCC lattices. A first way to reduce
dimension is by considering layered lattices, and minimize either among
competitors presenting different sequences of repetitions of the layers, or
among competitors presenting different shifts of the layers with respect to
each other. The second case presents the problem of minimizing theta functions
also on translated lattices, namely minimizing . Another way to reduce dimension is by considering
lattices with a product structure or by successively minimizing over concentric
layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics
questions, which we study in detail in two dimensions.Comment: 45 pages. 7 figure
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