STRUCTURAL INSTABILITY INDUCED BY NEAR SINGULAR MATRICES IN LATTICE-BASED CRYPTOGRAPHY

Abstract

Lattice-Based Cryptography (LBC) emerged in response to Shor’s algorithm, which demonstrated that quantum computers could break classical cryptographic systems (Peter Shor, 1994; 1997). LBC derives its security from the computational hardness of geometric problems such as the Shortest Vector Problem (SVP). These hardness assumptions are intrinsically tied to lattice geometry, determinant scaling, Gram–Schmidt orthogonal (GSO) behavior. Determinants define the volume of the fundamental parallelepiped and influence the density of lattice points. This paper establishes that near-singular matrices act as structural instability agents within lattice constructions. The degree of structural instability discloses the efficiency of the underlying vectors. We show that near-singularity induces the tendency of determinant collapse, Gram–Schmidt norm imbalance, condition number and Root Harmit Factor explosion. Near singularity effects on the lattice bases expose the strength of the dependency level of the vectors. Linear independence is a strict measure for lattice bases validity (Micciancio, 2002). Linear independence is a continuous phenomenon that affects numerical conditioning, motivating the need for stability measures in lattice representation (Peikert, 2016). When lattice bases are dependent or tend to be dependent, it increases reduction susceptibility under Lenstra–Lenstra–Lovász and Block Korkine–Zolotarev algorithms. The results show that the action of a near-singular matrix on a lattice base is a good indicator that measures the dependency level of vectors in a given lattice at the representation stage. It is not merely a numerical preference but a fundamental security check and necessity. 

Keywords: near-singular matrix, lattice base cryptography, LLL/BKZ reduction, determinant, Gram Schmidt Orthogonal, Hermit Factor, condition number.