The symmetry point of a unimodular lattice's secrecy function
Let $L \subset \mathbb{R}^n$ be a full-rank unimodular lattice, that is, integral ($\langle x, z \rangle \in \mathbb{Z}$ for $x, z \in L$) with covolume one. For $y > 0$ let $$\theta_L(y) = \sum_{x \in L} e^{-\pi y \|x\|^2}, \qquad \Xi_L(y) = \frac{\theta_{\mathbb{Z}^n}(y)}{\theta_L(y)}.$$ The supremum of the secrecy function $\Xi_L$ is the secrecy gain of $L$, introduced by Belfiore and Oggier for Gaussian wiretap coding, and Jacobi's formula gives $\Xi_L(1/y) = \Xi_L(y)$, as Belfiore and Solé observed. Ernvall-Hytönen proved that $\Xi_L$ attains its maximum at $y = 1$ for all extremal even unimodular lattices known in 2012, and gave a method to check this for any given unimodular lattice. For $\ell$-modular lattices the analogous statement, with the maximum at $y = 1/\sqrt{\ell}$, fails: Ernvall-Hytönen and Sethuraman showed that the 4-modular lattice $\mathbb{Z} \oplus \sqrt{2}\,\mathbb{Z} \oplus 2\mathbb{Z}$ is a counterexample.
Problem. Is $\Xi_L(y) \le \Xi_L(1)$ for every $n$, every such $L$ and every $y > 0$? The maximum need not be attained only at $y = 1$; for $L = \mathbb{Z}^n$ the function $\Xi_L$ is constant.
Source: Belfiore and Solé (2010), Conjecture 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NT — Number Theory
- Secondary
- IT, MG
- Keywords
- AIM Open Applied Problems, theta series, unimodular lattices, wiretap codes
References
@inproceedings{belfiore2010unimodular,
author = {Belfiore, Jean-Claude and Sol{\'e}, Patrick},
title = {Unimodular lattices for the {G}aussian wiretap channel},
booktitle = {2010 IEEE Information Theory Workshop},
address = {Dublin},
year = {2010},
pages = {1--5},
doi = {10.1109/CIG.2010.5592923},
eprint = {1007.0449},
archivePrefix = {arXiv},
}
@inproceedings{belfiore2010secrecy,
author = {Belfiore, Jean-Claude and Oggier, Fr{\'e}d{\'e}rique},
title = {Secrecy gain: a wiretap lattice code design},
booktitle = {2010 International Symposium on Information Theory and its Applications (ISITA)},
address = {Taichung},
year = {2010},
pages = {174--178},
doi = {10.1109/ISITA.2010.5650095},
}
@article{ernvallhytonen2012,
author = {Ernvall-Hyt{\"o}nen, Anne-Maria},
title = {On a conjecture by {B}elfiore and {S}ol{\'e} on some lattices},
journal = {IEEE Trans. Inform. Theory},
volume = {58},
number = {9},
year = {2012},
pages = {5950--5955},
doi = {10.1109/TIT.2012.2201915},
eprint = {1104.3739},
archivePrefix = {arXiv},
}
@article{ernvallhytonen2016,
author = {Ernvall-Hyt{\"o}nen, Anne-Maria and Sethuraman, B. A.},
title = {Counterexample to the generalized {B}elfiore--{S}ol{\'e} secrecy function conjecture for $l$-modular lattices},
journal = {IEEE Trans. Inform. Theory},
volume = {62},
number = {8},
year = {2016},
pages = {4514--4522},
doi = {10.1109/TIT.2016.2573826},
eprint = {1409.3188},
archivePrefix = {arXiv},
}
@misc{bollauf2024,
author = {Bollauf, Maiara F. and Lin, Hsuan-Yin},
title = {On the maximum flatness factor over unimodular lattices},
year = {2024},
eprint = {2403.16932},
archivePrefix = {arXiv},
note = {v1--v2 (2024) titled ``On the maximum theta series over unimodular lattices''; v3, 2025},
}
@misc{aim-belfiore-sole,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The symmetry point of a unimodular lattice's secrecy function}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/603-belfiore-sole.md},
note = {Entry 603 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00332, "The symmetry point of a unimodular lattice's secrecy function", clopen.solutions.