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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
PubMed
Summary
This summary is machine-generated.

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.

Keywords:
(n,k)-simpleHNFrandom integer latticerejection sampling method

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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.