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Full-Text Articles in Physical Sciences and Mathematics
Permutation Invariant Lattices, Lenny Fukshansky, Stephan Ramon Garcia, Xun Sun
Permutation Invariant Lattices, Lenny Fukshansky, Stephan Ramon Garcia, Xun Sun
CMC Faculty Publications and Research
We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ ∈ Sn, we study properties of the set of all lattices closed under the action of τ: we call such lattices τ-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio in [8, 9], which we previously studied in [1]. Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the …
On The Geometry Of Cyclic Lattices, Lenny Fukshansky, Xun Sun
On The Geometry Of Cyclic Lattices, Lenny Fukshansky, Xun Sun
CMC Faculty Publications and Research
Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest 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. Our main result is a counting estimate for the number of well-rounded …