Discrete Gaussian measures and new bounds of the smoothing parameter for lattices
In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, a simple form of uncertainty principle for discrete Gaussian measure is formulated. In the second part of the paper we prove two new bounds for the smoothing parameter of...
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| Vydáno v: | Applicable algebra in engineering, communication and computing Ročník 32; číslo 5; s. 637 - 650 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.11.2021
Springer Nature B.V |
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| ISSN: | 0938-1279, 1432-0622 |
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| Abstract | In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, a simple form of uncertainty principle for discrete Gaussian measure is formulated. In the second part of the paper we prove two new bounds for the smoothing parameter of lattices. Under the natural assumption that
ε
is suitably small, we obtain two estimations of the smoothing parameter:
η
ε
(
Z
)
≤
ln
(
ε
44
+
2
ε
)
π
.
This is a practically useful case. For this case, our upper bound is very close to the exact value of
η
ε
(
Z
)
in that
ln
(
ε
44
+
2
ε
)
π
-
η
ε
(
Z
)
≤
ε
2
552
.
For a lattice
L
⊂
R
n
of dimension
n
,
η
ε
(
L
)
≤
ln
(
n
-
1
+
2
n
ε
)
π
bl
~
(
L
)
. |
|---|---|
| AbstractList | In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, a simple form of uncertainty principle for discrete Gaussian measure is formulated. In the second part of the paper we prove two new bounds for the smoothing parameter of lattices. Under the natural assumption that ε is suitably small, we obtain two estimations of the smoothing parameter: ηε(Z)≤ln(ε44+2ε)π. This is a practically useful case. For this case, our upper bound is very close to the exact value of ηε(Z) in that ln(ε44+2ε)π-ηε(Z)≤ε2552.For a lattice L⊂Rn of dimension n, ηε(L)≤ln(n-1+2nε)πbl~(L). In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, a simple form of uncertainty principle for discrete Gaussian measure is formulated. In the second part of the paper we prove two new bounds for the smoothing parameter of lattices. Under the natural assumption that ε is suitably small, we obtain two estimations of the smoothing parameter: η ε ( Z ) ≤ ln ( ε 44 + 2 ε ) π . This is a practically useful case. For this case, our upper bound is very close to the exact value of η ε ( Z ) in that ln ( ε 44 + 2 ε ) π - η ε ( Z ) ≤ ε 2 552 . For a lattice L ⊂ R n of dimension n , η ε ( L ) ≤ ln ( n - 1 + 2 n ε ) π bl ~ ( L ) . |
| Author | Zhao, Chunhuan Xu, Guangwu Zheng, Zhongxiang |
| Author_xml | – sequence: 1 givenname: Zhongxiang surname: Zheng fullname: Zheng, Zhongxiang organization: Institute for Advanced Study, Tsinghua University – sequence: 2 givenname: Chunhuan surname: Zhao fullname: Zhao, Chunhuan organization: Institute for Advanced Study, Tsinghua University – sequence: 3 givenname: Guangwu surname: Xu fullname: Xu, Guangwu email: gxu4uwm@uwm.edu organization: Key Laboratory of Cryptologic Technology and Information Security of Ministry of Education, School of Cyber Science and Technology, Shandong University, Department of EE & CS, University of Wisconsin-Milwaukee |
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| Keywords | 11T71 Lattices Discrete Gaussian measure Smoothing parameter 11H06 94A60 Lattice based cryptography |
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| References | Banaszczyk (CR3) 1996; 16 Micciancio, Regev (CR9) 2007; 37 Tian, Liu, Xu (CR17) 2014; 142 Peikert (CR12) 2010; 2010 CR5 CR7 Stein, Shakarchi (CR15) 2003 CR18 Banaszczyk (CR2) 1995; 13 Cai (CR4) 2003; 126 CR16 CR14 Micciancio, Walter (CR10) 2017; 2017 CR13 CR11 Banaszczyk (CR1) 1993; 296 Donoho, Stark (CR6) 1989; 49 Micciancio, Goldwasser (CR8) 2002 W Banaszczyk (417_CR3) 1996; 16 D Micciancio (417_CR8) 2002 W Banaszczyk (417_CR1) 1993; 296 D Donoho (417_CR6) 1989; 49 D Micciancio (417_CR9) 2007; 37 W Banaszczyk (417_CR2) 1995; 13 E Stein (417_CR15) 2003 417_CR14 417_CR5 417_CR13 D Micciancio (417_CR10) 2017; 2017 C Tian (417_CR17) 2014; 142 417_CR7 417_CR11 J Cai (417_CR4) 2003; 126 C Peikert (417_CR12) 2010; 2010 417_CR18 417_CR16 |
| References_xml | – ident: CR18 – year: 2003 ident: CR15 publication-title: Fourier Analysis—An Introduction – volume: 296 start-page: 625 issue: 4 year: 1993 end-page: 635 ident: CR1 article-title: New bounds in some transference theorems in the geometry of numbers publication-title: Mathematische Annalen doi: 10.1007/BF01445125 – ident: CR14 – volume: 126 start-page: 9 issue: 1 year: 2003 end-page: 31 ident: CR4 article-title: A new transference theorem in the geometry of numbers and new bounds for Ajtai’s connection factor publication-title: Discrete Appl. Math. doi: 10.1016/S0166-218X(02)00216-0 – ident: CR16 – ident: CR13 – volume: 16 start-page: 305 year: 1996 end-page: 311 ident: CR3 article-title: Inequalities for convex bodies and polar reciprocal lattices in II: application of k-convexity publication-title: Discrete Comput. Geom. doi: 10.1007/BF02711514 – ident: CR11 – volume: 37 start-page: 267 issue: 1 year: 2007 end-page: 302 ident: CR9 article-title: Worst-case to average-case reductions based on Gaussian measures publication-title: SIAM J. Comput. doi: 10.1137/S0097539705447360 – volume: 13 start-page: 217 year: 1995 end-page: 231 ident: CR2 article-title: Inequalites for convex bodies and polar reciprocal lattices in publication-title: Discrete Comput. Geom. doi: 10.1007/BF02574039 – ident: CR5 – ident: CR7 – volume: 142 start-page: 47 year: 2014 end-page: 57 ident: CR17 article-title: Measure inequalities and the transference theorem in the geometry of numbers publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-2013-11744-2 – year: 2002 ident: CR8 publication-title: Complexity of Lattice Problems: A Cryptographic Perspective doi: 10.1007/978-1-4615-0897-7 – volume: 2017 start-page: 455 year: 2017 end-page: 485 ident: CR10 article-title: Gaussian sampling over the integers: efficient, generic, constant-time publication-title: Proc. CRYPTO – volume: 2010 start-page: 80 year: 2010 end-page: 97 ident: CR12 article-title: An efficient and parallel Gaussian sampler for lattices publication-title: Proc. CRYPTO – volume: 49 start-page: 906 year: 1989 end-page: 931 ident: CR6 article-title: Uncertainty principles and signal recovery publication-title: SIAM J. Appl. Math. doi: 10.1137/0149053 – ident: 417_CR5 doi: 10.1109/CCC.2013.31 – volume: 37 start-page: 267 issue: 1 year: 2007 ident: 417_CR9 publication-title: SIAM J. Comput. doi: 10.1137/S0097539705447360 – volume-title: Fourier Analysis—An Introduction year: 2003 ident: 417_CR15 – ident: 417_CR7 doi: 10.1145/1374376.1374407 – volume-title: Complexity of Lattice Problems: A Cryptographic Perspective year: 2002 ident: 417_CR8 doi: 10.1007/978-1-4615-0897-7 – volume: 142 start-page: 47 year: 2014 ident: 417_CR17 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-2013-11744-2 – ident: 417_CR11 doi: 10.1109/CCC.2007.12 – volume: 2017 start-page: 455 year: 2017 ident: 417_CR10 publication-title: Proc. CRYPTO – volume: 126 start-page: 9 issue: 1 year: 2003 ident: 417_CR4 publication-title: Discrete Appl. Math. doi: 10.1016/S0166-218X(02)00216-0 – volume: 49 start-page: 906 year: 1989 ident: 417_CR6 publication-title: SIAM J. Appl. Math. doi: 10.1137/0149053 – volume: 16 start-page: 305 year: 1996 ident: 417_CR3 publication-title: Discrete Comput. Geom. doi: 10.1007/BF02711514 – volume: 296 start-page: 625 issue: 4 year: 1993 ident: 417_CR1 publication-title: Mathematische Annalen doi: 10.1007/BF01445125 – ident: 417_CR13 doi: 10.1007/978-3-662-44709-3_20 – volume: 13 start-page: 217 year: 1995 ident: 417_CR2 publication-title: Discrete Comput. Geom. doi: 10.1007/BF02574039 – ident: 417_CR14 – ident: 417_CR18 – volume: 2010 start-page: 80 year: 2010 ident: 417_CR12 publication-title: Proc. CRYPTO – ident: 417_CR16 |
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| SubjectTerms | Artificial Intelligence Computer Hardware Computer Science Lattices (mathematics) Original Paper Parameters Smoothing Symbolic and Algebraic Manipulation Theory of Computation Uncertainty principles Upper bounds |
| Title | Discrete Gaussian measures and new bounds of the smoothing parameter for lattices |
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