Secrecy Gain of Formally Unimodular Lattices from Codes over the Integers Modulo 4
| Autor: | Bollauf, Maiara F., Lin, Hsuan-Yin, Ytrehus, Øyvind |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | Recently, a design criterion depending on a lattice's volume and theta series, called the secrecy gain, was proposed to quantify the secrecy-goodness of the applied lattice code for the Gaussian wiretap channel. To address the secrecy gain of Construction $\text{A}_4$ lattices from formally self-dual $\mathbb{Z}_4$-linear codes, i.e., codes for which the symmetrized weight enumerator (swe) coincides with the swe of its dual, we present new constructions of $\mathbb{Z}_4$-linear codes which are formally self-dual with respect to the swe. For even lengths, formally self-dual $\mathbb{Z}_4$-linear codes are constructed from nested binary codes and double circulant matrices. For odd lengths, a novel construction called odd extension from double circulant codes is proposed. Moreover, the concepts of Type I/II formally self-dual codes/unimodular lattices are introduced. Next, we derive the theta series of the formally unimodular lattices obtained by Construction $\text{A}_4$ from formally self-dual $\mathbb{Z}_4$-linear codes and describe a universal approach to determine their secrecy gains. The secrecy gain of Construction $\text{A}_4$ formally unimodular lattices obtained from formally self-dual $\mathbb{Z}_4$-linear codes is investigated, both for even and odd dimensions. Numerical evidence shows that for some parameters, Construction $\text{A}_4$ lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Results concerning the flatness factor, another security criterion widely considered in the Gaussian wiretap channel, are also discussed. Comment: arXiv admin note: text overlap with arXiv:2202.09236; Paper accepted for publication in IEEE Transactions on Information Theory |
| Databáze: | arXiv |
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