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@article{208741,
        author = {Pratiksha Nade and Rahul Mali and Sushma Chalke},
        title = {Linear Algebra Under Attack: A Historical Analysis of Hill Cipher Countermeasures and The Turn Toward Lattice-Based Hardness},
        journal = {International Journal of Innovative Research in Technology},
        year = {2026},
        volume = {13},
        number = {no},
        pages = {631-634},
        issn = {2349-6002},
        url = {https://ijirt.org/article?manuscript=208741},
        abstract = {Linear algebra supplied cryptography's first matrix cipher and, decades later, the lattice-based constructions built to resist quantum attacks. This paper traces the documented history of the Hill Cipher, from Lester Hill's 1929 transformation through a sequence of “secure” modifications permutation-based, circulant-matrix, eigenvalue-driven, and multi-key chaining schemes and shows that several of the most confidently claimed countermeasures were later broken by cryptanalysis exploiting the same underlying algebraic reversibility, including a 2025 attack that defeated a 2019 scheme within six years. We argue this pattern is not incidental: any transformation expressible as a solvable system of equations over a ring remains vulnerable once an attacker can reconstruct that system. We contrast this with Learning with Errors (LWE), which achieves lasting resistance through a formal reduction to worst-case lattice problems rather than structural complexity, and close by examining 2025 findings showing that even LWE's theoretical hardness estimates outpace concrete, implementation-level security benchmarking. This paper assembles that lineage, spanning 1929 to 2025, and shows that it follows a consistent pattern: countermeasures built on added linear-algebraic structure tend to be eventually broken because they remain solvable systems of equations. We then examine why LWE represents a genuine departure, and ask whether the same lesson is being applied proactively to today's post-quantum standards.},
        keywords = {Linear Algebra, Hill Cipher, Known-Plaintext Attack, Cryptanalysis, Learning with Errors, Lattice Cryptography, Post-Quantum Security.},
        month = {September},
        }