The Automorphism Group Of 48-dimensional And 72-dimensional Lattice

Lattice theory has attracted much attention in modern cryptography.With the deepening of the study of the theory of lattices,we begin to further understand and study the automorphism of lattices.Different types of lattices have different types of automorphisms,and to find the automorphism of lattice in several specific dimensions become the main topic presently.By discussing whether we can construct the automorphism group of the lattice,we can solve the shortest vector problem and the nearest vector problem in the lattice,so we are able to construct a lattice with higher dimension and find some automorphisms with special lattices which is profound.In this paper,we summarize the properties of lattices and the algebraic structures,and then,based on the related literatures and after that we get some understanding of the basic construction methods of 24-dimensional automorphisms,we try to explore the automorphism groups of higher-dimensional lattices and find a feasible way to construct the automorphism group.At the same time,taking full advantage of the characteristics of lattices and lattices,this paper studies the special properties of automorphism groups of 48-dimensional and 72-dimensional lattice.The content of this article is organized as follows:The first chapter is introduction.It tells the background and significance of the lattice,and introduces the research status of the authomorphism of the lattice theory,and concludes with a brief description of the main contents of this article.The second chapter is preparedness.It introduced the basic algebra and basic lattice theory that used in this paper.This knowledge provides theoretical support for the automorphism of subsequent research lattices.The third chapter is the main content.The examples and construction methods of48-dimensional single-mode even lattices are given,as well as a construction method of 72-dimensional automorphism,and it also gives the corresponding proofs.The fourth chapter is summary and prospect.The main content of this article is summarized,and the application of automorphism of the lattice has raised some new problems.