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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Measure inequalities and the transference theorem in the geometry of numbers


Authors: Chengliang Tian, Mingjie Liu and Guangwu Xu
Journal: Proc. Amer. Math. Soc. 142 (2014), 47-57
MSC (2010): Primary 06D50, 11H06; Secondary 03G10, 52C07
Published electronically: September 18, 2013
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Abstract: The measure inequalities of Banaszczyk have been important tools in applying discrete Gaussian measure over lattices to lattice-based cryptography. This paper presents an improvement of Banaszczyk's inequalities and provides a concise and transparent proof. This paper also generalizes the transference theorem of Cai to general convex bodies. The bound is better than that obtained by simply generalizing the $ l_2$ norm using the canonical norm inequalities.


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Additional Information

Chengliang Tian
Affiliation: Key Lab of Cryptologic Technology and Information Security, Ministry of Education, Shandong University, Jinan, 250100, People’s Republic of China – and – School of Mathematics, Shandong University, Jinan, 250100, People’s Republic of China
Address at time of publication: SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, People’s Republic of China
Email: chengliangtian@mail.sdu.edu.cn

Mingjie Liu
Affiliation: Institute for Advanced Study, Tsinghua University, Beijing, 100084, People’s Republic of China
Email: liu-mj07@mails.tsinghua.edu.cn

Guangwu Xu
Affiliation: Department of Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
Email: gxu4uwm@uwm.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11744-2
PII: S 0002-9939(2013)11744-2
Keywords: Convex body, lattice-based cryptography, Gaussian measures, transference theorem
Received by editor(s): August 6, 2011
Received by editor(s) in revised form: February 2, 2012, and February 26, 2012
Published electronically: September 18, 2013
Additional Notes: The first author was supported by the National Natural Science Foundation of China (Grant No. 61133013 and No. 60931160442)
The second author was supported by Tsinghua University Initiative Scientific Research Program No. 2009THZ01002.
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.