Perrin, D., & Voloch, J. F. (2024). Ordinary Isogeny Graphs with Level Structure. arXiv preprint arXiv:2411.02732v1.
This paper investigates the impact of adding level structures (specifically Γ0(N), Γ1(N), and Γ(N)) to ordinary components (volcanoes) of ℓ-isogeny graphs, focusing on determining the crater size and the number of components in these modified graphs.
The authors employ a mathematical approach, leveraging the theory of complex multiplication and class field theory. They analyze the action of generalized ideal class groups on sets of elliptic curves with specific endomorphism rings and level structures.
By considering generalized ideal class group actions, the authors successfully address the question of determining the crater size and the number of components in ordinary isogeny graphs enhanced with level structures. This provides a deeper understanding of the structural properties of these graphs.
This research contributes significantly to the theoretical understanding of isogeny graphs, which are fundamental objects in isogeny-based cryptography. The insights gained from analyzing level structures have implications for the design and analysis of cryptographic protocols based on isogenies.
The paper primarily focuses on the mathematical aspects of isogeny graphs with level structures. Further research could explore the practical implications of these findings for isogeny-based cryptography, such as analyzing the security and efficiency of cryptographic schemes employing these structures.
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