Contents

• Contents
• List of Tables
• List of Figures
• Introduction
  • Point Lattices
  • Reduced Lattice Bases
  • Lattice Basis Reduction Algorithms
• The Seysen Basis Reduction Algorithm
  • Theoretical Analysis
    • Sufficiency of $T_{i,j}^\lambda$ Matrices
    • Choosing Vector Pairs to Reduce
    • The S(A) Function
    • Choosing $\lambda$ Values
  • Empirical Analysis
    • Lazy vs. Greedy Selection Methods
    • Choosing $\lambda$ Values
    • Testing the $B_{\theta }$ Lattice
    • Testing Random Integer Lattices
  • When Seysen's Algorithm Fails
    • Row Moves Involving Three or Four Vectors
    • Simulated Annealing and Rapid Quenching
    • Using Hadamard Matrices to Permute Lattice Bases
  • Extending Seysen's Algorithm
    • General n-vector Row Operations
    • Alternate Selection Criteria
    • Alternate Choices of $\lambda$
    • Alternate S(A) Functions
• Solving Subset Sum Problems
  • Introduction
  • Theoretical Bounds on Solving Subset Sum Problems
  • Previous Empirical Methods
  • Using Seysen's Algorithm to Solve Subset Sum Problems
  • Empirical Tests Using Algorithm SL
• Conclusions
  • Candidate Lattices for Seysen Reduction
  • Modifying Algorithm SL
• Bibliography