Lattice Reduction for Modules, or How to Reduce ModuleSVP to ModuleSVP

We show how to generalize lattice reduction algorithms to module lattices. Specifically, we reduce $\gamma$-approximate ModuleSVP over module lattices with rank $k \geq2$ to $\gamma\u27$-approximate ModuleSVP over module lattices with rank $2 \leq \beta \leq k$. To do so, we modify the celebrated slide-reduction algorithm of Gama and Nguyen to work with module filtrations, a high-dimensional generalization of the ($\Z$-)basis of a lattice. The particular value of $\gamma$ that we achieve depends on the underlying number field $K$, the order $R \subseteq \mathcal{O}_K$, and the embedding (as well as, of course, $k$, $\beta$, and $\gamma\u27$). However, for reasonable choices of these parameters, the resulting value of $\gamma$ is surprisingly close to the one achieved by ``plain\u27\u27 lattice reduction algorithms, which require an arbitrary SVP oracle in the same dimension. In other words, we show that ModuleSVP oracles are nearly as useful as SVP oracles for solving higher-rank instances of approximate ModuleSVP. Our result generalizes the recent independent result of Lee, Pellet-Mary, Stehlé, and Wallet, which works in the important special case when $\beta = 2$ and $R = \mathcal{O}_K$ is the ring of integers of $K$ under the canonical embedding. Our reduction works for any $\beta$ dividing $k$, as well as arbitrary orders $R \subseteq \mathcal{O}_K$ and a larger class of embeddings. Indeed, at a high level our reduction can be thought of as a generalization of theirs in roughly the same way that block reduction generalizes LLL reduction

Short bases of lattices over number fields

Abstract. Lattices over number elds arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of Z-lattices, the choice of a nice, short (pseudo)-basis is important in many applications. In this article, we provide the rst algorithm that computes such a short (pseudo)-basis. We utilize the LLL algorithm for Z-lattices together with the Bosma-Pohst-Cohen Hermite Normal Form and some size reduction technique to nd a pseudo-basis where each basis vector belongs to the lattice and the product of the norms of the basis vectors is bounded by the lattice determinant, up to a multiplicative factor that is a eld invariant. As it runs in polynomial time, this provides an e ective variant of Minkowski's second theorem for lattices over number elds.

Inventário arbóreo em dois bairros paulistanos, Jardim da Saúde e Vila Vera, localizados na subprefeitura de Ipiranga Arboreal inventory of two São Paulo city neighboorhoods (Jardim da Saude and Vila Vera) located in Ipiranga district zone

Foi realizado levantamento quantiqualitativo de vegetais de porte arbóreo em dois bairros paulistanos, Vila Vera e Jardim da Saúde, situados na região Sudeste de São Paulo, não distando muito entre si, porém com características de ocupação de uso do solo bastante distintas. No Jardim da Saúde, encontraram-se 1.033 exemplares de 72 espécies botânicas, cuja altura média de todas as árvores foi de 8,07 m. A Caesalpinea peltophoroides Benth. foi a espécie mais frequente, com 20,68%, seguida da Largestroemia indica L., com 7,48%. Predominavam árvores com mais de 8,50 m de altura, sendo o pior indicador de sanidade vegetal a infestação de cupins, com 8,33% do total. Apenas 23,33% tinham situações de permeabilidade do passeio suficiente, e 5,71% delas se encontravam com conduções de podas para desobstrução das redes aéreas; o rebaixamento das árvores aconteceu em 7,74% dos exemplares. Na Vila Vera havia 178 árvores pertencentes a 42 espécies botânicas, sendo a média da altura total de 6,31 m. A espécie mais abundante foi a Caesalpinea peltophoroides Benth., com 24,71%; a segunda seria a Ligustrum lucidum W.T. Aiton, com 17,24% do total. Em 42,70% das árvores, a altura foi inferior a 4,50 m, e a sanidade vegetal estava comprometida em 12,37% dos exemplares pela infestação de cupins. Somente 7,87% dos exemplares estavam em situações de permeabilidade suficiente, as conduções de poda para desobstrução de redes eram de 3,38% e as podas de rebaixamento, de 13,48%. Havia uma média de 16,85 m de afastamento entre árvores no Jardim da Saúde, enquanto na Vila Vera esse indicativo era de 38,68 m.<br>An arboreal qualitative and quantitative study was conducted in two closely located neighborhoods (Vila Vera and Jardim da Saúde) in the southeast region of São Paulo. In spite of geographical vicinity, both neighborhoods have very different land occupation characteristics. In Jardim da Saúde we found 1033 tree specimens, belonging to 72 distinct botanic species, with an average height of 8.07m. Caesalpinea peltophoroides Benth. was the most frequent species found (20,68% of the trees), followed by Largestroemia indica L. (with 7.48%) Most trees have a height of 8.50 m and above. The worst health threat was termite infestation which affected 8.33% of the specimens. Only 23.33% of the trees were planted in an area which enough surface permeability. 5.71% of the specimens had been pruned to prevent interference with electrical lines and crown-reducing pruning had been done in 7.74% of the trees. In Vila Vera we found limited space conditions in regards to the width of the sidewalk and also in regards to the predominant type of site utilization. We counted 178 trees belonging to 42 distinct botanic species, with average height of 6.31m. The most common species is Caesalpinea peltophoroides Benth. (which accounted for 24.71% of the specimens), followed by a Ligustrum lucidum W.T. Aiton (with 17.24%). In this neighborhood 42.70% of the trees had a height of less than 4.5m, 12.37% were in poor health due to termite infestation. Only 7.87% of the specimens where planted in an area which enough surface permeability, while 16.85% where located in totally paved areas. 3.38% had been pruned to avoid interference with electrical lines and 13.48% had been crown-reducing pruned. In Jardim da Saúde the average distance between trees was 16.85m, while in Vila Vera it was 38.68m. In other words, the average distance between trees is about 2.29 greater in Vila Vera than in Jardim da Saúde

An LLL Algorithm for Module Lattices

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