Abstract
Let \(X \in {{\mathbb Z}}^{n \times m}\), with each entry independently and identically distributed from an integer Gaussian distribution. We consider the orthogonal lattice \(\varLambda ^\perp (X)\) of X, i.e., the set of vectors \(\mathbf {v}\in {{\mathbb Z}}^m\) such that \(X \mathbf {v}= \mathbf {0}\). In this work, we prove probabilistic upper bounds on the smoothing parameter and the \((m-n)\)-th minimum of \(\varLambda ^\perp (X)\). These bounds improve and the techniques build upon prior works of Agrawal et al. (Adv Cryptol 2013:97–116, 2013), and of Aggarwal and Regev (Chic J Theor Comput Sci 7:1–11, 2016).
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Notes
Note that an equivalent description of the distribution for X would be \(X \leftarrow (D_{{{\mathbb Z}},s})^{n \times m}\). Our choice follows prior works.
We recall that \(\eta _{\varepsilon }({{\mathbb Z}}^n) = {\mathcal {O}}(\sqrt{ \ln (n/\varepsilon )})\) (see Sect. 2).
The statistical distance between two distributions X and Y is half their \(\ell _1\)-distance, i.e., \(\varDelta (X,Y):=\frac{1}{2}\left\Vert X-Y\right\Vert _1 = \frac{1}{2} \sum _{\omega \in \varOmega } |X(\omega ) - Y(\omega )|\).
In fact, the following stronger result is proved in [13]: the number of primes in the interval \((x-x^{\alpha },x)\) is at least \(\frac{x^\alpha }{\log x}\) for \(\alpha < 7/12\). To simplify our statements, we use a looser bound.
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Acknowledgements
This work has been supported by ERC Starting Grant ERC-2013-StG-335086-LATTAC and by the European Union PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701). Most of the work leading to this article was done while the first and last authors were at ENS de Lyon.
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Kirshanova, E., Nguyen, H., Stehlé, D. et al. On the smoothing parameter and last minimum of random orthogonal lattices. Des. Codes Cryptogr. 88, 931–950 (2020). https://doi.org/10.1007/s10623-020-00719-w
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DOI: https://doi.org/10.1007/s10623-020-00719-w