Golden-Ratio Quasicrystalline Lattices in Worst-Case Lattice Cryptography — E8 Intelligence Research

FINDING: Lattice-based cryptography's security rests on the worst-case hardness of SVP/CVP in high-dimensional lattices, with quasicrystalline cut-and-project lattices (Z[φ] modules) forming a special structured subclass whose geometry is governed by the golden ratio. | MATH: Lattice L = {Σᵢ aᵢbᵢ : aᵢ ∈ ℤ}, basis B ∈ ℝ^(n×m). SVP: find v ∈ L\{0} minimizing ‖v‖₂. GapSVP_γ: decide if λ₁(L) ≤ 1 or > γ. Ajtai '96: worst-case GapSVP reduces to average-case SIS/LWE. Regev '05: LWE quantum-reduces to GapSVP_γ for γ = Õ(n). Bennett et al. (2023): 2^(n/2)-time SVP lower bound under SETH. Cut-and-project: L_∥ = π_∥(Z^d ∩ strip), with Z[φ] = {a + bφ : a,b ∈ ℤ}, φ = (1+√5)/2. | CONNECTION: Z[φ] is the ring of integers of ℚ(√5), the golden-ratio quadratic field. Its unit group is generated by φ (norm −1) and φ² (norm +1). The 1D Fibonacci quasicrystal is the cut-and-project of Z[φ] onto a line of slope 1/φ; the 2D Penrose tiling arises from the ℤ[φ]-module ℤ[φ]² projected from ℤ⁵ (or ℤ⁴ with the st Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com