Основные понятия
Zn achieves the largest theta series over unimodular lattices.
Аннотация
This article explores the maximum theta series over unimodular lattices, focusing on Zn as achieving the largest value. It discusses connections to physical layer security and cryptography, presenting technical contributions related to secrecy ratios and conjectures for unimodular lattices. The Belfiore-Solé and Regev and Stephens-Davidowitz conjectures are analyzed in relation to the theta series of lattices.
Introduction:
Physical layer security in communication.
Wiretap channel concept.
Secrecy rate definition.
Cryptography Perspective:
Smoothing parameter definition.
Reverse Minkowski theorem implications.
Global maximum theta series over stable lattices.
Flatness Factor and Unimodular Lattices:
Definitions of flatness factor and smoothing parameter.
Integral Regev and Stephens-Davidowitz Conjecture.
Scaled Construction A integral lattice theorem.
Relation between Belfiore-Solé and Regev and Stephens-Davidowitz Conjectures:
U-shaped function concept.
Theorem on U-shaped property for secrecy ratio minimization.
Sufficient condition for secrecy ratio minimization.
Secrecy Ratio of Construction A Unimodular Lattices:
Lemma on secrecy ratio expression for self-dual codes.
Theorem on Belfiore-Solé conjecture verification conditions.
Example with a binary self-dual code weight enumerator distribution.
Necessary Conditions for Regev and Stephens-Davidowitz Conjecture:
Weight distribution considerations for self-dual codes.
Verification techniques for Belfiore-Solé conjecture.
Статистика
Znはunimodular格子の最大シータ級数を達成します。