TY - GEN

T1 - Towards Strong Reverse Minkowski-Type Inequalities for Lattices

AU - Dadush, Daniel

AU - Regev, Oded

PY - 2016/12/14

Y1 - 2016/12/14

N2 - We present a natural reverse Minkowski-type inequality for lattices, which gives upper bounds on the number of lattice points in a Euclidean ball in terms of sublattice determinants, and conjecture its optimal form. The conjecture exhibits a surprising wealth of connections to various areas in mathematics and computer science, including a conjecture motivated by integer programming by Kannan and Lovasz (Annals of Math. 1988), a question from additive combinatorics asked by Green, a question on Brownian motions asked by Saloff-Coste (Colloq. Math. 2010), a theorem by Milman and Pisier from convex geometry (Ann. Probab. 1987), worst-case to average-case reductions in lattice-based cryptography, and more. We present these connections, provide evidence for the conjecture, and discuss possible approaches towards a proof. Our main technical contribution is in proving that our conjecture implies the l2 case of the Kannan and Lovasz conjecture. The proof relies on a novel convex relaxation for the covering radius, and a rounding procedure based on 'uncrossing' lattice subspaces.

AB - We present a natural reverse Minkowski-type inequality for lattices, which gives upper bounds on the number of lattice points in a Euclidean ball in terms of sublattice determinants, and conjecture its optimal form. The conjecture exhibits a surprising wealth of connections to various areas in mathematics and computer science, including a conjecture motivated by integer programming by Kannan and Lovasz (Annals of Math. 1988), a question from additive combinatorics asked by Green, a question on Brownian motions asked by Saloff-Coste (Colloq. Math. 2010), a theorem by Milman and Pisier from convex geometry (Ann. Probab. 1987), worst-case to average-case reductions in lattice-based cryptography, and more. We present these connections, provide evidence for the conjecture, and discuss possible approaches towards a proof. Our main technical contribution is in proving that our conjecture implies the l2 case of the Kannan and Lovasz conjecture. The proof relies on a novel convex relaxation for the covering radius, and a rounding procedure based on 'uncrossing' lattice subspaces.

KW - Geometry

KW - Lattices

KW - Minkowski's first theorem

UR - http://www.scopus.com/inward/record.url?scp=85009376175&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009376175&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2016.55

DO - 10.1109/FOCS.2016.55

M3 - Conference contribution

AN - SCOPUS:85009376175

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 447

EP - 456

BT - Proceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016

PB - IEEE Computer Society

T2 - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016

Y2 - 9 October 2016 through 11 October 2016

ER -