Physical Sciences and Mathematics | Open Access Articles

Lattices From Hermitian Function Fields, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj Jan 2016

Lattices From Hermitian Function Fields, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

Pomona Faculty Publications and Research

We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of …

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Permutation Invariant Lattices, Lenny Fukshansky, Stephan Ramon Garcia, Xun Sun Jan 2015

Permutation Invariant Lattices, Lenny Fukshansky, Stephan Ramon Garcia, Xun Sun

Pomona 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 S_n, i.e., if the lattice is closed under the action of some non-identity elements of S_n. Given a fixed element T E S_n, we study properties of the set of all lattices closed under the action of T: we call such lattices T-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio in [7,8], which we previously studied in [1]. Continuing our investigation, we discuss some basic properties of …

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Permutation Invariant Lattices, Lenny Fukshansky, Stephan Ramon Garcia, Xun Sun Jan 2015

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 …

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Lattices From Elliptic Curves Over Finite Fields, Lenny Fukshansky, Hiren Maharaj Jul 2014

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.

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Well-Rounded Zeta-Function Of Planar Arithmetic Lattices, Lenny Fukshansky Jan 2014

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 …

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On The Geometry Of Cyclic Lattices, Lenny Fukshansky, Xun Sun Jan 2014

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 …

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On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky Aug 2008

On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky

CMC Faculty Publications and Research

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of Z², as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum …

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On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky Jun 2008