Related Experiment Video
Updated: Jun 27, 2026
09:32
Stable DNA Motifs, 1D and 2D Nanostructures Constructed from Small Circular DNA Molecules
Published on: April 12, 2019
On (n,k)-Simple Random Integer Lattices.
Gengran Hu1,2
1School of Cyberspace, Hangzhou Dianzi University, Hangzhou 310018, China.
Entropy (Basel, Switzerland)
|June 26, 2026
Summary
This study introduces (n,k)-simple random integer lattices, bridging general lattices and simple Hermite Normal Forms (HNF). A new generation algorithm offers efficient construction for structured random lattices.
Area of Science:
- Number Theory
- Cryptography
- Discrete Mathematics
Background:
- Random integer lattices are crucial in lattice-based cryptography and algorithmic number theory.
- A 2016 model introduced unrestricted random integer lattices, with ~44% found in simple Hermite Normal Form (HNF).
Purpose of the Study:
- To bridge the gap between general random integer lattices and those in simple HNF.
- Introduce the (n,k)-simple random integer lattice model.
- Provide a theoretical and practical framework for structured random lattices.
Main Methods:
- Derivation of the asymptotic counting formula for (n,k)-simple random integer lattices.
- Computation of their density among all integer lattices.
- Development of a generation algorithm using rejection and inverse sampling.
Main Results:
- The asymptotic counting formula and density for (n,k)-simple random integer lattices are derived.
- A generation algorithm with O(n^2) expected running time is proposed.
- The concept of (n,k)-simple lattices provides controlled HNF structures.
Conclusions:
- The (n,k)-simple random integer lattice model effectively bridges existing lattice types.
- The proposed generation algorithm is efficient for practical applications.
- This work establishes a foundation for constructing structured random lattices with specific HNF properties.
Related Concept Videos
02:33Lattice Centering and Coordination Number
The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
Imagine taking a large number of identical...
Types of Unit Cells
Imagine taking a large number of identical...
01:12Bewley Lattice Diagram
The Bewley lattice diagram, developed by L. V. Bewley, effectively organizes the reflections occurring during transmission-line transients. It visually represents how voltage waves propagate and reflect within a transmission line, making it easier to understand the complex interactions that occur.
01:09Random Variables
A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the...
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the...
02:22Structures of Solids
Solids in which the atoms, ions, or molecules are arranged in a definite repeating pattern are known as crystalline solids. Metals and ionic compounds typically form ordered, crystalline solids. A crystalline solid has a precise melting temperature because each atom or molecule of the same type is held in place with the same forces or energy. Amorphous solids or non-crystalline solids (or, sometimes, glasses) which lack an ordered internal structure and are randomly arranged. Substances that...
01:27Lattice Energies of Ionic Crystals
Lattice energy represents the energy released when gaseous cations and anions combine to form an ionic solid, reflecting the strength of electrostatic interactions within the crystal. This process is fundamentally governed by Coulombic attraction between oppositely charged ions, where the potential energy varies inversely with the interionic distance and directly with the product of ionic charges. As ions approach one another, the electrostatic energy becomes increasingly negative, indicating a...
01:09Random Sampling Method
Sampling is a technique to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population. The sampling method ensures that samples are drawn without bias and accurately represent the population. Because measuring the entire population in a study is not practical, researchers use samples to represent the population of interest. Among the various sampling methods used by...