Breaking Neural Networks Through Algebraic Lens: A Preliminary Analysis of Mathematical Vulnerabilities in AI Architectures

Modern deep neural networks, particularly transformer architectures, have achieved unprecedented performance across various domains through sophisticated mathematical structures that remain incompletely understood from a security perspective. This paper presents a comprehensive framework for identifying and analyzing algebraic vulnerabilities in contemporary AI architectures through computational algebra and discrete mathematics. We demonstrate that the composition of linear transformations, non-linear activation functions, and attention mechanisms creates exploitable polynomial systems that can be systematically analyzed using advanced algebraic techniques. Our methodology reveals fundamental weaknesses in popular architectures including Vision Transformers (ViTs), BERT-based models, and Residual Networks (ResNets), showing how adversaries can construct attacks that exploit polynomial-time algorithms while acknowledging current computational limitations. Through rigorous mathematical analysis and experimental validation on networks up to 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> parameters, we establish theoretical bounds for these vulnerabilities and demonstrate attack success rates of 52-78% on individual network components. We provide detailed end-to-end attack demonstrations showing measurable prediction changes, with attack success defined as achieving targeted modifications in network outputs that exceed 10% deviation from original predictions. While computational scalability remains a significant constraint, the findings suggest that algebraic structure analysis provides crucial insights into AI system security that require consideration from both cryptographic and machine learning communities. We position this work as contributing to both mathematical interpretability of neural networks and security vulnerability analysis, with implications for developing algebraically-robust architectures.