APPLICATIONS OF ARITHMETIC GEOMETRY IN CRYPTOGRAPHY AND COMPUTATIONAL MATHEMATICS: ELLIPTIC CURVES, FINITE FIELDS AND EMERGING DIRECTIONS
Keywords:
Arithmetic Geometry, Elliptic Curve Cryptography, Post-Quantum Cryptography, Supersingular Isogenies, Perfectoid Spaces, Prismatic Cohomology, Algebraic Geometry Codes, Arithmetic StatisticsAbstract
Arithmetic geometry represents the profound synthesis of algebraic geometry and number theory, emerging as a foundational pillar for both abstract mathematical theory and applied computational science. This report provides an exhaustive investigation into the applications of arithmetic geometry within modern cryptography and computational mathematics, tracking the discipline's evolution from the classical study of Diophantine equations to the frontiers of post-quantum cryptographic infrastructure. The analysis extensively evaluates the integration of elliptic curves over finite fields, emphasizing the critical role of rational points, the Schoof-Elkies-Atkin algorithm, and the Elliptic Curve Discrete Logarithm Problem (ECDLP) in establishing secure communication frameworks. Furthermore, the report critically examines the rapid paradigm shift in isogeny-based cryptography, detailing the catastrophic Castryck-Decru attack on the Supersingular Isogeny Diffie-Hellman (SIDH) protocol—a vulnerability exposed through the arithmetic geometry of abelian surfaces and Richelot isogenies—and the subsequent pivot toward the SQIsign protocol. Beyond cryptography, the analysis explores the deployment of algebraic geometry codes on modular curves that surpass the Gilbert-Varshamov bound. Finally, the report synthesizes emerging theoretical developments shaping the future of computational mathematics, including perfectoid spaces, prismatic cohomology, derived algebraic geometry, and arithmetic statistics, illuminating how the most abstract geometrical frameworks continuously inform and revolutionize applied algorithmic paradigms.
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