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Abstract <p>The creation of cryptographic systems based on lattice theory is a promising line in the field of post-quantum cryptography. The aim of this work was to obtain new properties of lattices through related objects: dense packings of equal spheres. We propose a way of constructing lattice packings of equal spheres corresponding to the packing density of the Lambda series in dimensions 1–24, using a series of coefficients to the height of the fundamental parallelepiped of dimension <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\((n-1)\)</EquationSource> <!--CMatCMGU2570005Lyalin-m1--> </InlineEquation>: <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m2--> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/3\)</EquationSource> <!--CMatCMGU2570005Lyalin-m3--> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m4--> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(0\)</EquationSource> <!--CMatCMGU2570005Lyalin-m5--> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m6--> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/3\)</EquationSource> <!--CMatCMGU2570005Lyalin-m7--> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m8--> </InlineEquation>, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sqrt{-1}\)</EquationSource> <!--CMatCMGU2570005Lyalin-m9--> </InlineEquation>, <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m10--> </InlineEquation>, <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/3\)</EquationSource> <!--CMatCMGU2570005Lyalin-m11--> </InlineEquation>, <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m12--> </InlineEquation>, <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(0\)</EquationSource> <!--CMatCMGU2570005Lyalin-m13--> </InlineEquation>, <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m14--> </InlineEquation>, <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/3\)</EquationSource> <!--CMatCMGU2570005Lyalin-m15--> </InlineEquation>, <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m16--> </InlineEquation>, <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sqrt{-1}\)</EquationSource> <!--CMatCMGU2570005Lyalin-m17--> </InlineEquation>, <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m18--> </InlineEquation>, <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/3\)</EquationSource> <!--CMatCMGU2570005Lyalin-m19--> </InlineEquation>, <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m20--> </InlineEquation>, <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(0\)</EquationSource> <!--CMatCMGU2570005Lyalin-m21--> </InlineEquation>, <InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m22--> </InlineEquation>, <InlineEquation ID="IEq23"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/3\)</EquationSource> <!--CMatCMGU2570005Lyalin-m23--> </InlineEquation>, <InlineEquation ID="IEq24"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5153_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--CMatCMGU2570005Lyalin-m24--> </InlineEquation>. The construction of lattice packings of equal spheres using this procedure was used up to dimension 11 inclusive.</p>

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A Way of Constructing Lattice Packings of Equal Spheres Corresponding to the Packing Density of the Lambda Series

  • 期刊论文

  • 发表日期：

    2025年6月16日

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  • M. A. Lyalin,
  • S. A. Fomin

摘要

Abstract

The creation of cryptographic systems based on lattice theory is a promising line in the field of post-quantum cryptography. The aim of this work was to obtain new properties of lattices through related objects: dense packings of equal spheres. We propose a way of constructing lattice packings of equal spheres corresponding to the packing density of the Lambda series in dimensions 1–24, using a series of coefficients to the height of the fundamental parallelepiped of dimension \((n-1)\) : \(1/2\) , \(1/3\) , \(1/2\) , \(0\) , \(1/2\) , \(1/3\) , \(1/2\) , \(\sqrt{-1}\) , \(1/2\) , \(1/3\) , \(1/2\) , \(0\) , \(1/2\) , \(1/3\) , \(1/2\) , \(\sqrt{-1}\) , \(1/2\) , \(1/3\) , \(1/2\) , \(0\) , \(1/2\) , \(1/3\) , \(1/2\) . The construction of lattice packings of equal spheres using this procedure was used up to dimension 11 inclusive.