For over four decades, the security of global digital communication has rested almost entirely on the perceived hardness of two number-theoretic problems: integer factorization and the discrete logarithm problem. However, the advent of quantum computation—and specifically the efficacy of Shor's algorithm in solving the Hidden Subgroup Problem for finite abelian groups—has rendered these foundations fundamentally fragile. To preserve privacy in a post-quantum world, the mathematical community has been forced to look "beyond factorization" toward algebraic structures that lack the periodicity exploited by quantum Fourier transforms.
This talk provides a rigorous survey of the mathematical primitives underpinning the next generation of cryptographic standards. We begin by examining the geometry of lattices, transitioning from classical Gaussian reduction to the contemporary complexity of the Shortest Vector Problem (SVP) and the Learning with Errors (LWE) framework. We will analyze why adding "noise" to linear systems transforms a trivial linear algebra problem into an NP-hard geometric challenge that remains resistant even to quantum adversaries.
Furthermore, we will explore algebraic coding theory as a cryptographic resource, focusing on the HQC cryptosystem and the hardness of decoding general linear codes. We will also briefly discuss the information-theoretic security of hash-based structures and the role of one-way functions. By bridging the gap between abstract complexity theory and the 2026 landscape of standardized PQC, this session aims to highlight the deep mathematical beauty and the urgent practical necessity of these resilient structures.
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