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Articles 1 - 3 of 3
Full-Text Articles in Physical Sciences and Mathematics
Lattices From Elliptic Curves Over Finite Fields, Lenny Fukshansky, Hiren Maharaj
Lattices From Elliptic Curves Over Finite Fields, Lenny Fukshansky, Hiren Maharaj
CMC Faculty Publications and Research
In their well known book Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities.
Well-Rounded Zeta-Function Of Planar Arithmetic Lattices, Lenny Fukshansky
Well-Rounded Zeta-Function Of Planar Arithmetic Lattices, Lenny Fukshansky
CMC Faculty Publications and Research
We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at s=1 with a real pole of order 2, improving upon a result of Stefan Kühnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less than or equal to N is O(N log N) as N → ∞. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence …
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 …