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ON THE GEOMETRY OF CYCLIC LATTICES
"... Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D. Micciancio in [9] and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen [1 ..."
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Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D. Micciancio in [9] and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen [12] showed that on cyclic lattices of prime dimension N, the short independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices, in particular proving that SVP is in fact equivalent to SIVP on a positive proportion of cyclic lattices in every dimension N. On the other hand, we also show that for a positive proportion of cyclic lattices in every dimension the two problems are different. To conclude, we introduce a class of sublattices of ZN closed under the action of subgroups of the permutation group SN, which are a natural generalization of cyclic lattices, and show that our results extend to all such lattices closed under the action of an N-cycle.
Creating a Challenge for Ideal Lattices
"... Abstract. Lattice-based cryptography is one of the candidates in the area of post-quantum cryptography. Cryptographic schemes with security reductions to hard lattice problems (like the Shortest Vector Problem SVP) offer an alternative to recent number theory-based schemes. In order to guarantee asy ..."
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Abstract. Lattice-based cryptography is one of the candidates in the area of post-quantum cryptography. Cryptographic schemes with security reductions to hard lattice problems (like the Shortest Vector Problem SVP) offer an alternative to recent number theory-based schemes. In order to guarantee asymptotic efficiency, most lattice-based schemes are instantiated using polynomial rings over integers. These lattices are called ideal lattices. It is assumed that the hardness of lattice problems in lattices over integer rings remains the same as in regular lattices. In order to prove or disprove this assumption, we instantiate random ideal lattices that allow to test algorithms that solve SVP and its approximate version. The Ideal Lattice Challenge allows online submission of short vectors to enter a hall of fame for full comparison. We adjoin a set of first experiments and a first comparison of ideal and regular lattices.
Natural Density Distribution of Hermite Normal Forms of Integer Matrices ∗
, 2011
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