A new transference theorem in the geometry of numbers and new bounds for Ajtai's connection factor

Authors:
Jin-Yi Cai
Affiliations:
Department of Computer Science and Engineering, State University of New York, Bell Hall, Buffalo, NY
Venue:
Discrete Applied Mathematics - Special issue: Special issue devoted to the fifth annual international computing and combinatories conference (COCOON'99) Tokyo, Japan 26-28 July 1999
Year:
2003

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Abstract

We prove a new transference theorem in the geometry of numbers, giving optimal bounds relating the successive minima of a lattice with the minimal length of generating vectors of its dual. It generalizes the transference theorem due to Banaszczyk. We also prove a stronger bound for the special class of lattices possessing nε-unique shortest lattice vectors. The theorem imply consequent improvement of the Ajtai connection factors in the connection of average-case to worst-case complexity of the shortest lattice vector problem. Our proofs are non-constructive, based on discrete Fourier transform.