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Efficient implementation of isogeny-based cryptography

par Stopar, Miha
Président du jury Markowitch, Olivier
Promoteur Petit, Christophe
Publication Non publié, 2026-01-28

Thèse de doctorat

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Résumé :

Isogeny-based cryptography is one of the candidates for post-quantum public-key cryptography, built on the arithmetic and geometry of elliptic curves, abelian varieties, and their isogenies. This thesis introduces the mathematical and algorithmic background necessary to place these constructions in context and presents several practical implementations developed as part of recent research.The first part develops the geometric background, covering elliptic curves, higher-dimensional abelian varieties, and the isogenies linking them. While the main research contributions of the thesis are largely driven by implementation, optimization, and experimental evaluation, this theoretical material is presented first to develop the mathematical foundations of the area, reflecting the fact that a significant part of the doctoral work was devoted to developing a deeper understanding of the underlying theory in parallel with implementation efforts.The second part provides background on selected aspects of isogeny-based cryptography, focusing on protocols and algorithms.The third part contains the main research contributions of the thesis. It includes several publications co-authored during the PhD, each presenting new algorithms or constructions together with practical implementations and benchmarks. These works cover improved methods for computing isogenies, constructions based on radical isogenies and theta structures, and protocols arising from supersingular graphs and class-group actions. Together, they illustrate how isogeny-based cryptographic primitives can be implemented and evaluated in practice, and they provide empirical data that informs future design choices.