# American Institute of Mathematical Sciences

November  2015, 9(4): 391-413. doi: 10.3934/amc.2015.9.391

## Probability estimates for fading and wiretap channels from ideal class zeta functions

 1 Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076, Finland, Finland 2 Department of Mathematics and Statistics, University of Helsinki, FI-00014, Finland 3 Department of Electrical and Computer Systems Engineering, Monash University, P.O. Box 35, Clayton, Victoria 3800, Australia

Received  March 2013 Published  November 2015

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems.
Citation: David Karpuk, Anne-Maria Ernvall-Hytönen, Camilla Hollanti, Emanuele Viterbo. Probability estimates for fading and wiretap channels from ideal class zeta functions. Advances in Mathematics of Communications, 2015, 9 (4) : 391-413. doi: 10.3934/amc.2015.9.391
##### References:
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##### References:
 [1] , SAGE open source mathematics software system, http://www.sagemath.org/. [2] J.-C. Belfiore and F. Oggier, Lattice code design for the rayleigh fading wiretap channel, IEEE International Conference on Communications, (2011), 1-5. doi: 10.1109/iccw.2011.5963544. [3] J.-C. Belfiore and F. Oggier, An error probability approach to mimo wiretap channels, IEEE Trans. on Comm., 61 (2013), 3396-3403. doi: 10.1109/TCOMM.2013.061913.120278. [4] J.-C. Belfiore and F. Oggier, Secrecy gain: A wiretap lattice code design, IEEE International Symposium on Information Theory and its Applications, (2010), 174-178. doi: 10.1109/ISITA.2010.5650095. [5] J.-C. Belfiore and P. Solé, Unimodular lattices for the gaussian wiretap channel, IEEE Information Theory Workshop, (2010), 1-5. doi: 10.1109/CIG.2010.5592923. [6] J. Ducoat and F. Oggier, An analysis of small dimensional fading wiretap lattice codes, IEEE International Symposium on Information Theory, (2014), 966-970. doi: 10.1109/ISIT.2014.6874976. [7] A.-M. Ernvall-Hytönen and C. Hollanti, On the Eavesdropper's Correct Decision in Gaussian and Fading Wiretap Channels Using Lattice Codes, IEEE Information Theory Workshop, 2011. [8] C. Hollanti and E. Viterbo, Analysis on Wiretap Lattice Codes and Probability Bounds from Dedekind Zeta Functions, IEEE International Congress on Ultra-Modern Telecommunications and Control Systems and Workshops, 2011. [9] C. Hollanti, E. Viterbo and D. Karpuk, Nonasymptotic probability bounds for fading channels exploiting Dedekind zeta functions, preprint, arXiv:1303.3475. [10] S. Lang, Algebraic Number Theory, Second edition. Graduate Texts in Mathematics, 110. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0853-2. [11] S. Leung-Yan-Cheong and M. Hellman, The Gaussian wire-tap channel, IEEE Trans. on Inf. Theory, 24 (1978), 451-456. doi: 10.1109/TIT.1978.1055917. [12] F. Oggier, P. Solé and J.-C. Belfiore, Lattice codes for the wiretap gaussian channel: Construction and analysis, Information Theory, IEEE Tran, pp (2015), p1, arXiv:1103.4086 (2013). doi: 10.1109/TIT.2015.2494594. [13] F. Oggier and E. Viterbo, Algebraic number theory and code design for rayleigh fading channels, Foundations and Trends in Communications and Information Theory, 1 (2004), 333-416. doi: 10.1561/0100000003. [14] S. Ong and F. Oggier, Wiretap lattice codes from number fields with no small norm elements, Designs, Codes, and Cryptography, 73 (2014), 425-440. doi: 10.1007/s10623-014-9935-7. [15] R. Vehkalahti and H.-F. Lu, An algebraic look into MAC-DMT of lattice space-time codes, IEEE International Symposium on Information Theory, (2011), 2831-2835. doi: 10.1109/ISIT.2011.6034091. [16] R. Vehkalahti and H.-F. Lu, Diversity-multiplexing gain tradeoff: A tool in algebra?, IEEE Information Theory Workshop, (2011), 135-139. doi: 10.1109/ITW.2011.6089362. [17] R. Vehkalahti, H.-F. Lu and L. Luzzi, Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra, IEEE Trans. on Inf. Theory, 59 (2013), 6060-6082. doi: 10.1109/TIT.2013.2266396. [18] R. Vehkalahti and L. Luzzi, Connecting DMT of division algebra space-time codes and point counting in lie groups, IEEE International Symposium on Information Theory, (2012), 3038-3042. doi: 10.1109/ISIT.2012.6284119. [19] A. Wyner, The wire-tap channel, Bell Syst. Tech. Journal, 54 (1975), 1355-1387. doi: 10.1002/j.1538-7305.1975.tb02040.x.
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