×

zbMATH — the first resource for mathematics

Wiretap lattice codes from number fields with no small norm elements. (English) Zbl 1335.94106
Summary: We consider the problem of designing reliable and confidential lattice codes, known as wiretap lattice codes, for fast fading channels. We identify as code design criterion for finite lattice constellations a finite sum of inverse of algebraic norms, in fact the analogous for finite sums of the known criterion for infinite constellations, which motivates us to study totally real number fields with few elements of small norms, over which ideal lattices can be built to provide wiretap lattice codes. More precisely, we first narrow down our search to Abelian number fields known to provide reliable lattice codes, among which we look for no small inert primes and large regulators.

MSC:
94B60 Other types of codes
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11Y16 Number-theoretic algorithms; complexity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bayer-Fluckiger E., Oggier F., Viterbo E.: Algebraic lattice constellations: bounds on performance. IEEE Trans. Inf. Theory 52(1), 319-327 (2006). · Zbl 1182.94060
[2] Belfiore J.-C., Oggier F.: Lattice code design for the Rayleigh fading wiretap channel. In: Communications Workshops, International Conference on Communications (ICC), Kyoto (2011).
[3] Ebeling W.: Lattices and codes. A course partially based on lectures by Friedrich Hirzebruch. In: Advanced Lectures in Mathematics. Springer, Wiesbaden (2013). · Zbl 1257.11066
[4] Everest G.R., Loxton J.H.: Counting algebraic units with bounded height. J. Number Theory 44, 222-227 (1993). · Zbl 0786.11064
[5] Giraud X., Boutillon E., Belfiore J.-C.: Algebraic tools to build modulation schemes for fading channels. IEEE Trans. Inf. Theory 43(3), 938-952 (1997). · Zbl 0880.94015
[6] Hollanti C., Viterbo E.: Analysis on wiretap lattice codes and probability bounds from Dedekind zeta functions. In: ICUMT, Hungary (2011).
[7] Hollanti C., Viterbo E., Karpuk D.: Nonasymptotic probability bounds for fading channels exploiting Dedekind zeta functions. Preprint. http://arxiv.org/pdf/1303.3475 (2013).
[8] Lin F., Oggier F.: A classification of unimodular lattice wiretap codes in small dimensions. IEEE Trans. Inf. Theory 59(6). http://arxiv.org/pdf/1201.3688 (2013). · Zbl 1364.94761
[9] Ling C., Luzzi L., Belfiore J.-C., Stehlé D.: Semantically secure lattice codes for the Gaussian wiretap channel. Preprint. http://arxiv.org/abs/1210.6673 (2012). · Zbl 1360.94417
[10] Marcus D.A.: Number Fields. Springer, New York (1977). · Zbl 0383.12001
[11] Oggier F., Viterbo E.: Algebraic number theory and code design for Rayleigh fading channels. Found. Trends Commun. Inf. Theory 1(3), 333-415 (2004). · Zbl 1137.94014
[12] Oggier F., Solé P., Belfiore J.-C.: Lattice codes for the wiretap Gaussian channel: construction and analysis. Preprint. http://arxiv.org/abs/1103.4086v3 (2013).
[13] Ong S.S., Oggier F.: Lattices from totally real number fields with large regulator. In: WCC (2013).
[14] Vehkalahti R., Lu H.F., Luzzi L.: Inverse determinant sums and connections between fading channel information theory and algebra. IEEE Trans. Inf. Theory 59(9). http://arxiv.org/abs/1111.6289 (2013). · Zbl 1364.94472
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.